Abstract
Classical geometric and topological operations on polyhedra, maps and polytopes often give rise to structures with the same symmetry group as the original one, but with more flags. In this paper we introduce the notion of voltage operations on maniplexes, as a way to unify the study of such operations and generalize them to other geometric and combinatorial structures such as abstract polytopes, hypermaps, maniplexes or hypertopes. This can be done since our technique provides a way to study classical operations in a graph theoretic setting, and thus to apply a voltage operation one only needs that the combinatorial structure in hand can be understood as an nvalent properly nedge colored graph. For example, in the case of abstract polytopes, the partial order can be encoded into the socalled flag graph of the polytope and the voltage operation is therefore applied to such flag graph to then be recovered as a partial order. We focus on studying the interactions between voltage operations and the symmetries of the operated object, and show that these operations can be potentially used to build maniplexes with prescribed symmetry type graphs. Moreover, a complete characterization of when an operation can be seen as a voltage operation is given.
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1 Introduction
Traditionally, operations on polyhedra induce local transformations on the vertices, edges or faces. Classical examples include the Wythoffian ones, such as truncation and medial (see [3, chs. VIII and XI ] and Sect. 4). Classical operations often give rise to maps or polytopes with the same symmetry group as the original one, but with more flags. Thus, by applying one of these operations to a polytope we get a new polytope with more flag orbits. Furthermore, it is not difficult to see that, often, the “local configuration” of the flags does not depend on the original polytope but on the operation per se. Could we use operations to get any local configuration of the flag orbits? We shall explain this question more precisely later on.
Operations on maps and polytopes have been studied previously. For example, in [24], Orbanič, Pellicer and Weiss explore map operations to build korbit maps. They take a combinatorial approach and divide each flag (incident triplets vertexedgeface) of a map into several new ones. In a very recent paper [6] Cunningham, Pellicer and Williams introduce stratified operations to study monodromy groups of some important families of nonregular maniplexes (a generalization of the flag graph of polytopes, where some conditions are relaxed; see Sect. 2.2). This concept is closely related to our concept of voltage operation and many of their results can be written in our language and vice versa. However, not all classical operations have been studied in this combinatorial form. As an example, we have the snub operation, which is well known. The snub has the particularity that one should not take all flags of the original convex polyhedron \(\mathcal {P}\) to obtain the operated one, but only half of them. Alternatively, one can take all the flags of \(\mathcal {P}\) and obtain two connected components that, if \(\mathcal {P}\) is regular, one might regard as the two enantiomorphic forms of the snub. The previously studied operations assign one or more new flags to each flag of the operated object to obtain a connected object; voltage operations allow this to change, so we can obtain a much broader class of operations on polyhedra, maps, polytopes or maniplexes.
The flag graph of a polytope \(\mathcal {P}\) is a properly edgecolored nvalent graph whose vertices are the flags of \(\mathcal {P}\) and two of them are adjacent if the corresponding flags differ in exactly one element. The colors of the edges are given by the type of element they differ in. The quotient of the flag graph of \(\mathcal {P}\) by its symmetry group is the symmetry type graph of \(\mathcal {P}\). The symmetry type graph can be thought as a way to represent the local configuration of the flag orbits.
We can notice that when we apply a classical operation to different regular polyhedra, the resulting polyhedra often have isomorphic symmetry type graphs. (For example, if \({{\,\textrm{Med}\,}}\) represents the medial operation and \(\mathcal {P}\) is a regular polyhedron, the symmetry type graph of \({{\,\textrm{Med}\,}}(\mathcal {P})\) will have either one single vertex, or two vertices and an edge of color 2 joining them, depending on whether or not \(\mathcal {P}\) is selfdual.) Moreover, if we apply an operation to two polyhedra, both with the same symmetry type graph \(\mathcal {T}\), it is very likely that both resulting polyhedra have the same symmetry type graph \(\mathcal {T}'\).
A natural question to ask is: given a properly edgecolored nvalent graph \(\mathcal {T}\), is there a maniplex whose symmetry type graph is \(\mathcal {T}\)? Going back to operations, can we find an operation \({{\,\mathrm{\mathcal {O}}\,}}\) such that if \(\mathcal {P}\) is a regular maniplex, then the symmetry type graph of \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {P})\) is precisely \(\mathcal {T}\)?
These questions give rise to voltage operations as a potential useful tool to find answers. Voltage operations define a technique that uses voltage graphs to generalize the above mentioned classical operations, as well as many others, even in higher dimensions (or ranks). These operations can also be defined for maniplexes. But not only that, they can also be defined for premaniplexes, a more general concept that includes both maniplexes and their symmetry type graphs.
It is important to remark that voltage graphs have been used before in constructing polytopelike structures. Notably, in [26] the authors use voltage graphs to build 2orbit nmaniplexes (for \(n \ge 4\)) with prescribed symmetry type graphs. In [4] the authors use voltage graphs, without explicitly mentioning them, to find generators and relations for the automorphism group of 3orbit polytopes. In [14], the first and second authors of this manuscript exploited extensively the use of voltage graphs to solve some relevant problems in the area, namely, problems 1 and 2 in [5] (see also [22]).
We start this paper by giving some basic definitions that we shall need throughout the manuscript. In Sect. 3 we give the definition of a voltage operation and look in detail into the mix of two maniplexes (also called parallel product [31]) as a voltage operation; we also study some connectivity properties of the voltage operations. In Sect. 4 we explore some examples of classical operations on polytopes and polyhedra. In particular we show how pyramids, prisms, Wythoffian operations, among others, can be seen as voltage operations. In Sect. 5 we describe how automorphisms and voltage operations interact. We show that if \(\mathcal {M}\) is a maniplex with symmetry type \(\mathcal {T}\) and \({{\,\mathrm{\mathcal {O}}\,}}\) is a voltage operation, then every automorphism of \(\mathcal {M}\) induces an automorphism of \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {M})\). That is, the automorphism group of \(\mathcal {M}\) is a subgroup of the automorphism group of \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {M})\). Moreover, we show that the symmetry type graph of \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {M})\) with respect to the automorphism group of \(\mathcal {M}\) can be obtained by applying the voltage operation to \(\mathcal {T}\). In Sect. 6 we show that voltage operations are closed under composition and we find a simple way to describe the composition of two voltage operations. Finally, in Sect. 7 we give conditions for two voltage operations to be equivalent. We also characterize voltage operations in terms of what they do to the flag graph of the universal polytope \(\mathcal {U}\) (introduced in [27] and defined in Sect. 2.2 as a maniplex). More precisely, in Theorem 5.1 we prove that if \({{\,\mathrm{\mathcal {O}}\,}}\) is an operation that assigns a premaniplex \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {X})\) to each premaniplex \(\mathcal {X}\), then \({{\,\mathrm{\mathcal {O}}\,}}\) is a voltage operation if and only if \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {U}/\Gamma ) \cong {{\,\mathrm{\mathcal {O}}\,}}(\mathcal {U})/\Gamma \) for every group \(\Gamma \le \Gamma (\mathcal {U})\).
2 Preliminaries
2.1 Graphs
In this work we use the definition of graph used in [19], which is slightly more general than the usual definition. A graph X is a quadruple \((D,V,I,(\cdot )^{1})\) where D and V are disjoint sets, \(I:D \rightarrow V\) is a mapping and \((\cdot )^{1}: D \rightarrow D\) is an involutory permutation of D. The set V is the set of vertices of X, the set D is the set of darts of X. For a dart d, the vertex I(d) is the initial vertex or starting point of d and \(d^{1}\) is the inverse of d. The terminal vertex or endpoint of d is the starting point of \(d^{1}\). The edges are the orbits of D under the action of \((\cdot )^{1}\). If an edge consists only of a single dart, that is, a dart d that satisfies \(d^{1}=d\), then it is called a semiedge. A loop is an edge consisting of two darts whose initial vertex is the same. A link is an edge that is not a loop or a semiedge. That is, an edge whose two darts are different and have different starting points.
Usually a graph X is defined by its vertex set V(X) and the set E(X) of edges. We can recover the set of darts as the set of formal pairs
The function I is given by \(I({}^{e} v)=v\) and if an edge e is incident to the vertices u and v, then \(\left( {}^{e} v \right) ^{1} = {}^{e} u\). Formally speaking, this definition turns loops into semiedges. In this work we will not use graphs with loops hence we allow this small abuse.
If \(X=(V(X),D(X), I_{X}, (\cdot )^{1}_{X})\) and \(Y=(V(Y),D(Y), I_{Y}, (\cdot )^{1}_{Y})\) are graphs, a graph homomorphism \(f: X \rightarrow Y\) is a pair of functions \(f=(f_V, f_D)\) such that \(f_V: V(X) \rightarrow V(Y)\), \(f_D: D(X) \rightarrow D(Y)\) that satisfy
for every dart \(d \in D(X)\). In other words, the initial vertex (resp. inverse) of the image of a dart is the image of initial vertex (resp. inverse) of such dart. In this work we shall evaluate graph homomorphisms (hence, isomorphisms and automorphism, defined below) on the right, but any other function on the left. If \(v\in V(X)\) and \(d \in D(X)\), we shall write vf and df instead of \(vf_{V}\) and \(df_{D}\). If both \(f_{V}\) and \(f_{D}\) are bijective and \(f^{1}:=(f_{V}^{1},f_{D}^{1})\) is also a graph homomorphism, then we say that f (and \(f^{1})\) is an isomorphism and that X and Y are isomorphic (and write \(X \cong Y\)). Naturally, for a graph X, an automorphism is an isomorphism \(f:X \rightarrow X\).
A path is a finite sequence \(W=(d_{1}, \dots , d_{k})\) of darts such that the endpoint of \(d_{i}\) is the starting point of \(d_{i+1}\) for \(i \in \{0, \dots , k1\}\). We usually omit commas and parentheses and simply write \(W=d_{1} \dots d_{k}\). The number k is the length of W. The starting point of \(d_1\) is the starting point of W; we also say that W starts at \(I(d_{1})\). Similarly, the endpoint of W is the endpoint of \(d_{k}\) and we say that W ends at this vertex. We might slightly abuse notation and define a path of length 0 as a single vertex. A path is closed if its starting point and its endpoint are the same vertex.
Notions such as valency of a vertex, cycle, subgraphs, connectivity, connected components and trees extend naturally from the classic definition of graphs and paths.
Given a path \(W = d_{1} \dots d_{k}\), an elementary (graph) move consists in inserting or removing a pair of consecutive inverse darts at any point in the sequence \(d_{1}, \dots , d_{k}\). If a path \(W'\) can be obtained from a path W by applying a series of elementary moves then we say that the paths are graphhomotopic (and write \(W \sim W'\)). Clearly graphhomotopy is an equivalence relation and we often identify a path with its homotopy class. Observe that if W is graphhomotopic to a path of length 0 then W must be closed.
Given two paths \(W_1\) and \(W_2\) we say that they are compatible if the endpoint of \(W_1\) is the starting point of \(W_2\). We can operate compatible paths by concatenation. That is, if \(W_1=d_{1} \dots d_{k}\) and \(W_2=a_{1} \dots a_{\ell }\) then \(W_1W_2= d_{1} \dots d_{k} a_{1} \dots a_{\ell }\). If \(W_1 \sim W_1'\) and \(W_2 \sim W_2'\) then \(W_1W_2 \sim W_1'W_2'\), which implies that we can operate not only compatible paths but homotopy classes of compatible paths.
The fundamental groupoid of a graph X, denoted by \(\Pi (X)\) is the set of graphhomotopy classes of X with the partial operation defined above. If u is a vertex in X, then the fundamental group of X at u, denoted by \(\Pi ^{u}(X)\) is the set of graphhomotopy classes of closed paths at u with concatenation as operation. Observe that \(\Pi ^{u}(X)\) is actually a group and that if W is a path from v to u then \(\Pi ^{v}(X) = W \Pi ^{u}(X) W^{1} \).
If X is a graph and G is a group, a voltage assignment is a function \(\xi : \Pi (X) \rightarrow G\) that satisfies \(\xi (W_1 W_2) = \xi (W_2)\xi (W_1)\) for any two compatible paths \(W_1\) and \(W_2\).^{Footnote 1} The group G is called the voltage group of \(\xi \) and the pair \((X,\xi )\) is called a voltage graph.
Since every path can be thought as the product of its darts, we can regard a voltage assignment as a function \(\xi :D(X) \rightarrow G\) such that for every dart d, \(\xi (d^{1})= \xi (d)^{1}\). The voltage of a path \(W= d_{1}, \dots , d_{k}\) is simply \(\xi (d_{k}) \cdots \xi (d_{1})\).
If \((X,\xi )\) is a voltage graph, the derived graph is the graph \(X^{\xi }\) whose vertices and darts are the elements in \(V(X) \times G\) and \(D(X) \times G\), respectively. The initial vertex of a dart (d, g) is (I(d), g) and its inverse is \((d^{1}, \xi (d)g)\). The adjacent vertices of a given vertex (x, g) are the vertices \((y, \xi (d)g)\) for each dart d starting at x and ending at y. Equivalently, two vertices (x, g) and (y, h) are connected if there exists a dart d starting at x and ending at y whose voltage is \(h g^{1}\).
If X is a graph and \(\xi : \Pi (X) \rightarrow \Gamma \) and \(\zeta : \Pi (X) \rightarrow \Gamma \) are voltage assignments, then \(\xi \) and \(\zeta \) are equivalent if there is an isomorphism \(X^{\xi } \rightarrow X^{\zeta }\) such that the following diagram commutes:
where the arrows pointing to X are the projection to the first coordinate, which is a homomorphism.
The following theorem is well known (see [19]):
Theorem 2.1
If \((X,\xi )\) is a connected voltage graph and \(x_0\) is a vertex in X, there exists a voltage assignment \(\xi '\) on X satisfying that:

\(\xi '\) is equivalent to \(\xi \);

for every dart d in X, \(\xi '(d)\in \xi (\Pi ^{x_0}(X))\); and

there is a spanning tree T of X such that for every dart d in T, \(\xi '(d)=1\).
Note that if X is connected, without loss of generality, we can assume that a voltage assignment \(\xi \) on X satisfies the last two conditions of Theorem 2.1. We can think of the spanning tree T as a fundamental region for the group G.
2.2 Premaniplexes and Maniplexes
A properly nedgecolored graph is an nvalent graph X (that is, a graph whose vertices have all valency n) and a function \(c: D(x) \rightarrow \left\{ 0, \dots , n1 \right\} \) such that \(c(d) = c(d^{1})\) for every dart d and if \(d_{1}\) and \(d_{2}\) are such that \(I(d_{1}) = I(d_{2})\), then \(c(d_{1})\ne c(d_{2})\). Observe that c induces a proper edgecoloring in the classical sense.
A premaniplex of rank n or simply npremaniplex is a properly nedgecolored graph \(\mathcal {X}\) such that every vertex is the starting point of one dart of each color and if \(i,j \in \left\{ 0, \dots , n1 \right\} \) are such that \(\left ij \right \ge 2 \), then the alternating paths of length 4 with colors i, j are closed.
If a premaniplex \(\mathcal {X}\) is connected and simple, that is, if there are no semiedges or parallel edges, then we say that \(\mathcal {X}\) is a maniplex. Maniplexes were introduced in [32] as a combinatorial generalization of maps to higher ranks. Constructions of maniplexes with given symmetry properties can be found in [26]. Natural examples of maniplexes are the flag graphs of polytopes. In fact, if \(\mathcal {X}\) is a maniplex, we usually call its vertices flags. Moreover, if \(\mathcal {M}\) is an nmaniplex and \(i \in \{0, \dots , n1\}\), the ifaces of \(\mathcal {M}\) are the connected components of \(\mathcal {M}\) after removing the edges of color i. More generally, if \(0 \le k < \ell \le n1\) the \((k,\ell )\)sections of \(\mathcal {M}\) are the connected components of \(\mathcal {M}\) after removing the edges of color i if \(i < k\) or \(i > \ell \). Note that when \(\mathcal {M}\) is the flag graph of a polytope \(\mathcal {P}\), the faces and sections of \(\mathcal {M}\) are in correspondence with the faces and sections of \(\mathcal {P}\). This close relation between flag graphs of polytopes and maniplexes allows us to think of polytopes in a graphtheoretical approach. In this paper, we slightly abuse language and whenever we talk of a polytope \(\mathcal {P}\) we actually refer to the flag graph of \(\mathcal {P}\). In [9] GarzaVargas and Hubard characterize when a maniplex is the flag graph of a polytope. Finally, observe that the notions of flags, sections and faces extend naturally to premaniplexes. However, we shall keep the use of the word flag just for the case when a premaniplex is actually a maniplex.
If \(\mathcal {X}\) is a premaniplex, whenever we write \(x \in \mathcal {X}\) we mean that that x is a vertex in \(\mathcal {X}\). If x is a vertex in an npremaniplex, and \(i \in \left\{ 0, \dots , n1 \right\} \) we denote by \({}^{i}x\) the dart of color i whose starting point is x. We denote by \(x^{i}\) the iadjacent vertex of x, that is, the endpoint of the dart \({}^{i}x\).
Given two npremaniplexes \(\mathcal {X}\) and \(\mathcal {X}'\), a (premaniplex) homomorphism from \(\mathcal {X}\) to \(\mathcal {X}'\) is a function that preserves iadjacencies, for \(i\in \{0, \dots , n1\}\). In the particular case where \(\mathcal {X}\) and \(\mathcal {X}'\) are prepolytopes, these homomorphisms are called rapmaps. It is easy to prove that if \(\mathcal {X}'\) is connected, every homomorphism with codomain \(\mathcal {X}'\) is surjective (see [23, Lemma 2.5] for a proof for rapmaps that naturally extends to homomorphism of premaniplexes). In general, every connected component of \(\mathcal {X}'\) is either contained completely in the image of a homomorphism or in its complement. The class of all npremaniplexes together with the premaniplex homomorphisms as arrows forms a category which we will call \(\textrm{pMpx}^n\). A surjective homomorphism is called a covering. If there is a covering from \(\mathcal {X}\) to \(\mathcal {X}'\) we say that \(\mathcal {X}\) covers \(\mathcal {X}'\). It is easy to see that if \(\mathcal {X}'\) is connected, any homomorphism from \(\mathcal {X}\) to \(\mathcal {X}'\) is a covering. The notions of isomorphism and automorphism of premaniplexes are defined in the usual way. The automorphism group of \(\mathcal {X}\) is denoted by \(\Gamma (\mathcal {X})\), and acts naturally on vertices, ifaces and \((k,\ell )\)sections. Such actions will be considered as right actions. A premaniplex \(\mathcal {X}\) is regular (or reflexible) if \(\Gamma (\mathcal {X})\) acts transitively on vertices. We say that \(\mathcal {X}\) is a korbit premaniplex if \(\Gamma (\mathcal {X})\) induces k orbits on vertices. These notions coincide with the analogous notions for polytopes. For example, the flag graph of a polytope is a regular maniplex if and only if the polytope is regular itself.
If \(\mathcal {X}\) is an npremaniplex and \(\Gamma \le \Gamma (\mathcal {X})\) the quotient \(\mathcal {X}/\Gamma \) is the nedgecolored graph whose vertices are the orbits \(\{x\Gamma : x \in V(\mathcal {X})\}\) and for \(i \in \{0, \dots , n1\}\) \((x\Gamma )^{i} = (x^{i})\Gamma \). Observe that the adjacencies are welldefined since \((x^{i}) \gamma = (x \gamma )^{i}\) for every \(\gamma \in \Gamma (\mathcal {X})\). Moreover, it is straightforward that \(\mathcal {X}/\Gamma \) is an npremaniplex.
If \(\mathcal {M}\) is a maniplex and \(\Gamma \le \Gamma (\mathcal {M})\), the symmetry type graph of \(\mathcal {M}\) with respect to \(\Gamma \) is the quotient \(\mathcal {M}/\Gamma \). In particular, when \(\Gamma = \Gamma (\mathcal {M})\) then the quotient \(\mathcal {M}/\Gamma \) is the symmetry type graph of \(\mathcal {M}\) (see [4, 22]).
The universal rankn Coxeter group (with string diagram) is the group \(\mathcal {C}^{n}=\left\langle \rho _{0}, \dots \rho _{n1} \right\rangle \) defined by the following relations:
The universal nmaniplex \(\mathcal {U}^{n}\) is the Cayley graph associated with the universal Coxeter group \(\mathcal {C}^{n}\). That is, the vertex set of \(\mathcal {U}^{n}\) is \(\mathcal {C}^{n}\) and for \(\gamma \in \mathcal {C}^{n}\), the iadjacent vertex of \(\gamma \) is \(\rho _{i}\gamma \). The maniplex \(\mathcal {U}^{n}\) is in fact the flag graph of the universal polytope \(\left\{ \infty , \dots , \infty \right\} \) introduced by Schulte in [27] (see [20, sect. 3D] [12, Theorem 5.2]). We omit the rank of both the universal Coxeter group and the universal maniplex whenever it is implicit.
Since the universal Coxeter group \(\mathcal {C}\) acts transitively by automorphisms on the universal maniplex \(\mathcal {U}\), then \(\mathcal {U}\) is a regular maniplex. In fact, \(\Gamma (\mathcal {U})\) is precisely \(\mathcal {C}\).
We denote by \(r_{i}\) the permutation of \(V(\mathcal {U})\) that swaps every vertex x with \(x^{i}\). More precisely, for \(\gamma \in \mathcal {C}= V(\mathcal {U})\), \(r_i: \gamma \mapsto \rho _i\gamma \).
The monodromy group of \(\mathcal {U}\), denoted by \({{\,\textrm{Mon}\,}}(\mathcal {U})\), is the permutation group of \(V(\mathcal {U})\) generated by \(\{r_{0}, \dots , r_{n1}\}\). We call monodromies the elements of \({{\,\textrm{Mon}\,}}(\mathcal {U})\). The group \({{\,\textrm{Mon}\,}}(\mathcal {U})\) admits a (left) action on \(\mathcal {X}\) that is compatible with the (right) action of \(\Gamma (\mathcal {X})\), that is
for every \(x \in V(\mathcal {X})\), \(\gamma \in \Gamma (\mathcal {X})\) and \(i \in \{0,\dots , n1\}\).
Observe that the elements \(r_0, \dots , r_{n1}\) satisfy the relations
In fact, there exists an isomorphism from \(\mathcal {C}\) to \({{\,\textrm{Mon}\,}}(\mathcal {U})\) mapping \(\rho _{i}\) to \(r_{i}\) (see [23, Theorem 3.9]).
The relations in Equation (2.3) imply that if \(\mathcal {X}\) is any npremaniplex, then \({{\,\textrm{Mon}\,}}(\mathcal {U})\) acts on \(\mathcal {X}\) by \(r_i: x \mapsto x^{i}\) for every vertex x and \(i \in \{0, \dots , n1 \}\). The monodromy group of \(\mathcal {X}\) is the permutation group induced by this action. Naturally, we denote this group by \({{\,\textrm{Mon}\,}}(\mathcal {X})\). It is important to remark that the term connection group has been used for what we call the monodromy group and has gained popularity in the last few years.
Recall that for a vertex x and \(i \in \{0, \dots , n1\}\), \({}^{i}x\) denotes the dart of color i whose starting point is x. We can generalize this notation. If W is a path starting at x and following the colors \(i_{1}, i_{2}, \dots , i_{k}\) then we write \(W={}^{i_{k}, i_{k1}, \dots , i_{1}} x\). For every vertex x and \(i_{1}, i_{2}, \dots , i_{k} \in \left\{ 0, \dots , n1 \right\} \), the path \({}^{i_{k}, i_{k1}, \dots , i_{1}} x\) ends at the vertex \(r_{i_{k}} \cdots r_{i_{1}}x\).
Given a vertex y of \(\mathcal {U}\) and a path \(W={}^{i_k,i_{k1}\ldots ,i_1} x\) in a premaniplex \(\mathcal {X}\), the lift of W starting at y is simply the path \({}^{i_k,i_{k1},\ldots ,i_1} y\) in \(\mathcal {U}\).
An elementary (maniplex) move on a path \({}^{w} x\) with \(w=i_{k}, i_{k1}, \dots , i_{1}\) consists of either adding or removing the same color two times at any two consecutive positions (i.e. if \(v=i_k,\ldots ,i_\ell ,j,j,i_{\ell 1},\ldots ,i_1\) then \({}^{w} x\mapsto {}^{v} x\) and \({}^{v} x\mapsto {}^{w} x\) are elementary moves) or swapping two nonconsecutive colors in consecutive positions, more precisely, if \(i_\ell i_{\ell 1}>1\), and \(u=i_k,\ldots ,i_{\ell +1},i_{\ell 1},i_\ell ,i_{\ell 2},\ldots ,i_1\) then \({}^{w} x\mapsto {}^{u} x\) is an elementary move. We say that two paths in a premaniplex are maniplexhomotopic if we can turn one into the other by a finite sequence of elementary (maniplex) moves. Observe that two paths in an premaniplex are homotopic if their lifts to the universal maniplex \(\mathcal {U}\) starting at a the same vertex also end at the same vertex.
The homotopy class of a path in a premaniplex is uniquely determined by its starting vertex and a monodromy of the universal maniplex. If \(w \in {{\,\textrm{Mon}\,}}(\mathcal {U}^{n})\) and \(w=r_{i_{k}} \cdots r_{i_{1}}\) is a word representing w, then we denote by \({}^{w} x\) the homotopy class of the path \({}^{i_{k}, i_{k1}, \dots , i_{1}} x\). Observe that any two words representing w yield homotopic paths and if \(W_{1}, W_{2} \in {}^{w} x\), then both paths end at wx.
Let \(\mathcal {X}\) be a premaniplex and let \(\Pi (\mathcal {X})\) be its fundamental groupoid. If \(\xi :\Pi (\mathcal {X})\rightarrow \Gamma \) is a voltage assignment such that \(\xi (W)\) is the identity in \(\Gamma \) whenever W is a path of length 4 alternating between two nonconsecutive colors, we say that the pair \((\mathcal {X},\xi )\) is a voltage premaniplex. In other words, a voltage premaniplex is a voltage graph where the voltage of the maniplex homotopy class of a path is well defined.
3 Voltage operations
Let \(\mathcal {X}\) be an npremaniplex and let \(\mathcal {Y}\) be an mpremaniplex. Consider the voltage assignment \(\eta : \Pi (\mathcal {Y}) \rightarrow {{\,\textrm{Mon}\,}}(\mathcal {U}^{n})\). We define the properly medgecolored graph \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) in the following way: the vertex set of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is \({{\,\mathrm{{V}}\,}}(\mathcal {X}) \times {{\,\mathrm{{V}}\,}}(\mathcal {Y})\) and, for each \(i\in \{0,1,\dots , m\}\), there is an edge of color i from (x, y) to \(\left( \eta ({}^{i}y)x,r_i y\right) \) (see Fig. 1). Since, in \(\mathcal {Y}\), there is an iedge from y to \(r_iy\), then the darts \({}^{i}y\) and \({}^{i}r_iy\) are inverse, implying that \(\eta ({}^{i}r_iy)\eta ({}^{i}y)\) is the identity. Thus, each vertex of the graph \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) has exactly one iedge and, hence, it has valency m.
In the terminology of [19], \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is the derived graph from the voltage space \((\mathcal {X},{{\,\textrm{Mon}\,}}(\mathcal {U});\eta )\) with \(\mathcal {X}\) being the abstract fiber.
A pair \(\left( \mathcal {Y},\eta \right) \) is called a voltage operator if it is a voltage premaniplex with \(\eta : \Pi (\mathcal {Y}) \rightarrow {{\,\textrm{Mon}\,}}(\mathcal {U}^{n})\) (or, if the rank of \(\mathcal {Y}\) is m, an \(\left( n,m \right) \)voltage operator). Similarly, we say that the mapping \(\mathcal {X}\mapsto \mathcal {X}\rtimes _\eta \mathcal {Y}\), where \(\mathcal {X}\) runs over all npremaniplexes and \((\mathcal {Y},\eta )\) is a fixed voltage operator, is a voltage operation (or an (n, m)voltage operation).
Given premaniplexes \(\mathcal {X}\) and \(\mathcal {Y}\) and a voltage assignment \(\eta \), we shall see that \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is not only a properly medgecolored graph, but an mpremaniplex itself, although it need not be a maniplex (even if \(\mathcal {X}\) and/or \(\mathcal {Y}\) are maniplexes). Before showing this, we give some straightforward examples of voltage operations.
Let us denote by \({\textbf{1}}^{n}\) the symmetry type graph of a regular nmaniplex. That is, the premaniplex with only one vertex and n semiedges (see Fig. 2A). Moreover, if G is a group and \(\eta : {\textbf{1}}^{n} \rightarrow G\) is a voltage assignment that assigns \(g_{i}\) to the semiedge of color i, then we denote this voltage graph by \(\left( {\textbf{1}}^{n}, [g_{0}, \dots , g_{n1}] \right) \).
Example 3.1
The concept of a dautomorphism of a polytope is defined in [16], and can be generalized to maniplexes in a straightforward way. Let \({{\mathcal {M}}}\) be an nmaniplex and consider \({{\,\textrm{Mon}\,}}({{\mathcal {U}}}^n) =\langle r_0, r_1, \dots , r_{n1} \rangle \) and d an automorphism of \({{\,\textrm{Mon}\,}}(\mathcal {U}^n)\). We shall keep the vertices of \(\mathcal {M}\) and the action of \({{\,\textrm{Mon}\,}}(\mathcal {U}^n)\) on them, but we choose a new set of labeled generators of the same permutation group acting on the same set of flags to obtain a new maniplex, \({{\mathcal {M}}}^d\). More precisely, the maniplex \({{\mathcal {M}}}^d\) is defined as follows: the vertices of \(\mathcal {M}^d\) are the vertices of \(\mathcal {M}\) and given a vertex x, the dart of color i starting at x ends at \(d(r_i)x\). In other words, in \(\mathcal {M}^d\), \(x^i = d(r_i)x\). If \(\mathcal {M}\cong \mathcal {M}^d\), we call each isomorphism \(\varphi :\mathcal {M}\rightarrow \mathcal {M}^d\) a dautomorphism of \(\mathcal {M}\), and we say that \(\mathcal {M}\) is dautomorphic.
The maniplex \(\mathcal {M}^d\) can easily be seen as an (n, n)voltage operation:
if \((\mathcal {Y},\eta ) = ({\textbf{1}}^{n},[d(r_0), d(r_1) \dots , d(r_{n1})])\), then \(\mathcal {M}\rtimes _\eta \mathcal {Y}= \mathcal {M}^d\).
Classical examples of the above operation are the dual and the Petrial of an nmaniplex \(\mathcal {X}\):

If \((\mathcal {Y},\eta ) = ({\textbf{1}}^{n},[r_0, \dots , r_{n1}])\), then \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is \(\mathcal {X}\) itself.

If \((\mathcal {Y},\eta ) = ({\textbf{1}}^{n},[r_{n1}, \dots , r_{0}])\), then \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is the dual of \(\mathcal {X}\).

If \((\mathcal {Y},\eta ) = ({\textbf{1}}^{n},[r_0, \dots , r_{n4}, r_{n3}r_{n1}, r_{n2}, r_{n1}])\), then \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is the generalized Petrial of \(\mathcal {X}\).
In the above examples we know that whenever \(\mathcal {X}\) is in fact a maniplex, then so is \(\mathcal {X}\rtimes _\eta \mathcal {Y}\).
If \((\mathcal {Y}, \eta )\) is a voltage operator, we want to show that \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is a premaniplex, for every premaniplex \(\mathcal {X}\). Thus, we need to show that the alternating paths in \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) of length 4 with colors i and j, with \(\left ij \right \ge 2 \), are closed. In other words, given a premaniplex \(\mathcal {X}\) and a voltage operator \((\mathcal {Y}, \eta )\) we want \({}^{i,j,i,j} (x,y)=(x,y)\), whenever \(\left ij \right \ge 2 \), for every \(x\in \mathcal {X}\) and \(y \in \mathcal {Y}\).
More generally, we are interested in obtaining properties for the paths in \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) from the paths in \(\mathcal {X}\) or \(\mathcal {Y}\). The following result is a straightforward but useful observation towards this direction.
Remark 3.2
Let \((\mathcal {Y}, \eta )\) be a voltage operator. Assume that \(W={}^{i_{k},\dots , i_{1}} y\) is a path in \(\mathcal {Y}\) starting at y. For every \(x \in \mathcal {X}\) the path \({}^{i_{k},\dots , i_{1}} (x,y)\) in \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) that starts at (x, y) and follows the same colors as W finishes at \(\left( \eta (W)x, r_{i_{k}} \cdots r_{i_{1}}y\right) \).
By Remark 3.2 the path \({}^{i,j,i,j} (x,y)\) starts at (x, y) and finishes at \(\left( \eta ({}^{i,j,i,j} y)x, r_ir_jr_ir_j y\right) \). Thus, the path \({}^{i,j,i,j} (x,y)\) is closed if and only if
Note now that since \(\mathcal {Y}\) is a premaniplex, then \(r_{i}r_{j}r_{i}r_{j}y = y\) for every \(y\in \mathcal {Y}\), which implies that \({}^{i,j,i,j} (x,y)\) is closed if and only if \(\eta ({}^{i,j,i,j} y)\) fixes x. But \(\eta ({}^{i,j,i,j} y)\) is the identity, since \((\mathcal {Y}, \eta )\) is a voltage premaniplex. Therefore, we have the following proposition.
Proposition 3.3
Given a voltage premaniplex \((\mathcal {Y}, \eta )\) and a premaniplex \(\mathcal {X}\), the voltage operation \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is a premaniplex.
Since we are working with voltage premaniplexes, for the rest of the paper, whenever we refer to paths being “homotopic” or to the “homotopy class” of a path, we are thinking in terms of maniplex homotopy. In the same way, the notation \(\Pi (\mathcal {Y})\) will denote the fundamental groupoid consisting of paths in \(\mathcal {Y}\) considered up to maniplex homotopy.
Although we have special interest on voltage operations that give rise to connected premaniplexes, there are interesting examples of voltage operations in which we (often) obtain disconnected objects.
A rooted premaniplex is a pair \((\mathcal {X},x)\) where \(\mathcal {X}\) is a connected premaniplex and the root x is a vertex of \(\mathcal {X}\). If \(\mathcal {X}\) is not connected, then \((\mathcal {X},x)\) denotes the rooted premaniplex with the connected component of x in \(\mathcal {X}\) as its underlying premaniplex. If \((\mathcal {X},x)\) and \((\mathcal {Y},y)\) are rooted premaniplexes and \((\mathcal {Y},\eta )\) is a voltage operator, then \((\mathcal {X},x)\rtimes _\eta (\mathcal {Y},y)\) denotes the rooted premaniplex \((\mathcal {X}\rtimes _\eta \mathcal {Y}, (x,y))\).
Example 3.4
Let \(\mathcal {X}\) be a npremaniplex, \(1 \le k < \ell \le n\) and let \((\mathcal {Y},\eta ) = ({\textbf{1}}^{\ell  k  1},[r_{k+1}, \dots , r_{\ell 1}])\). Then \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is a graph whose connected components are the \((k,\ell )\)sections of \(\mathcal {X}\). In particular, if \(k=1\) and \(\mathcal {X}\) is a maniplex, then \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) determines the set of all \(\ell \)faces of \(\mathcal {X}\). Moreover, if y denotes the only vertex in \(\mathcal {Y}\), for each \(x \in \mathcal {X}\), then \((\mathcal {X},x)\rtimes _\eta (\mathcal {Y},y)\) is the \(\ell \)face of \(\mathcal {X}\) that contains x.
Note that if \(\mathcal {Y}\) is disconnected, then \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) consists of disjoint copies of \(\mathcal {X}\rtimes _{\eta _i}\mathcal {Y}_i\), where \(\mathcal {Y}_i\) runs over the connected components of \(\mathcal {Y}\) and \(\eta _i\) is the restriction of \(\eta \) to \(\Pi (\mathcal {Y}_i\)).
In a recent paper [6] the term stratified operations is introduced. They are operations on maniplexes defined in terms of a set A of strata admitting a left action of \({{\,\textrm{Mon}\,}}(\mathcal {U})\) and a function \(\varphi \) that assigns a monodromy to each pair \((a,r_i)\) where \(r_i\) is a generator of \({{\,\textrm{Mon}\,}}(\mathcal {U})\). We can describe stratified operations in terms of voltage operations. If \({{\,\mathrm{\mathcal {O}}\,}}\) is a stratified operation with strata set A, we can define the voltage operator \((\mathcal {Y},\eta )\) by taking the vertices of \(\mathcal {Y}\) to be the set A, defining the adjacencies by \(a^i=r_i a\), and defining the voltage assignment \(\eta \) by \(\eta ({}^{i}a)=\varphi (a,r_i)\). There may be a subtle difference between the stratified operation \({{\,\mathrm{\mathcal {O}}\,}}\) and the voltage operation defined by \((\mathcal {Y},\eta )\). Mainly, if \(\mathcal {M}\) is a maniplex \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {M})\) must be a maniplex too, and therefore connected, while \(\mathcal {M}\rtimes _\eta \mathcal {Y}\) might have more than one connected component. One can check that all stratified operations are the result of applying a voltage operation and then choosing one connected component. The converse, however is not true. There are voltage operations that do not define stratified operations when one chooses a connected component of the result. This is because the definition of stratified operation requires that the natural projection \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {M})\rightarrow \mathcal {M}\) must be surjective. The snub operation described in Sect. 4 and discussed further in Sect. 5 is a voltage operation, but not a stratified operation, since each connected component only uses half of the flags of the original maniplex when this is orientable.
3.1 The Mix as a Voltage Operation on Premaniplexes
The mix of two regular polytopes was defined in terms of their automorphism groups (see [21, 23]). In [5] the mix is defined for rooted maniplexes as a natural generalization of the parallel product of rooted maps (see [31]), and hence it is also defined for rooted polytopes. If \((\mathcal {M}, x)\) and \((\mathcal {N}, y)\) are rooted maniplexes, then \((\mathcal {M},x) \diamondsuit (\mathcal {N},y)\) is the smallest maniplex that covers both \((\mathcal {M}, x)\) and \( (\mathcal {N}, y)\). In particular, it is connected. We shall see that the mix can be defined as a voltage operation on premaniplexes.
Let \(\mathcal {Y}\) be an npremaniplex. We denote by \(\mu : \Pi (\mathcal {Y}) \rightarrow {{\,\textrm{Mon}\,}}(\mathcal {U}^{n})\) the voltage assignment that maps each dart of color i to \(r_{i}\). In other words \(\mu ({}^{\omega } y):=\omega \). We call \(\mu \) the mixing voltage (for \(\mathcal {Y}\)) and we say that \((\mathcal {Y},\mu )\) is a mix operator. Then \(\mathcal {X}\rtimes _\mu \mathcal {Y}\) is the mix \(\mathcal {X}\diamondsuit \mathcal {Y}\), and it is easy to see that this generalizes the same concept previously defined for regular abstract polytopes. By rooting the premaniplexes, this voltage operation generalizes the one for rooted maniplexes.
With our definition of the mix as a voltage operation we allow the resulting graph to be disconnected. However, each connected component of \(\mathcal {X}\rtimes _\mu \mathcal {Y}\) covers both \(\mathcal {X}\) and \(\mathcal {Y}\), so if at least one of them is simple (that is, a maniplex) the mentioned components are maniplexes as well. Moreover, the connected component of \(\mathcal {X}\diamondsuit \mathcal {Y}\) containing the vertex (x, y) is the smallest premaniplex that covers both \((\mathcal {X},x)\) and \((\mathcal {Y},y)\) (cf. [5, Proposition 3.10]).
We can use this definition of the mix to find the smallest cover of a maniplex satisfying some property, as we shall see in the next example.
Example 3.5
Let \(I\subset \{0,1,\ldots ,n1\}\). We denote by \({\textbf{2}}^n_I\) the npremaniplex with 2 vertices with semiedges of colors in I at each vertex and links of colors not in I between the 2 vertices (see Fig. 2B). We can use these premaniplexes to describe certain properties of maniplexes: orientable nmaniplexes (those that are bipartite) are those that cover \({\textbf{2}}^n_\emptyset \) and vertexbipartite maniplexes (those whose 1skeleton is bipartite) are those that cover \({\textbf{2}}^n_{\{1,2,\ldots ,n1\}}\), for example. In general, if \(\mathcal {X}\) covers \({\textbf{2}}^n_I\) it means that there is a coloring of the vertices (or flags) of \(\mathcal {X}\) with two colors such that iadjacent flags are of the same color if and only if \(i\in I\). See [18, 26] for a detailed discussion on flag colorings.
If a maniplex \(\mathcal {X}\) does not cover \({\textbf{2}}^n_I\), then \(\mathcal {X}\diamondsuit {\textbf{2}}^n_I\) is a maniplex that covers \(\mathcal {X}\) but also covers \({\textbf{2}}^n_I\). We call this the double cover of \(\mathcal {X}\) with respect to I. If \(\mathcal {X}\) is nonorientable, \(\mathcal {X}\diamondsuit {\textbf{2}}^n_\emptyset \) is the so called orientable double cover of \(\mathcal {X}\).
Note that if \(\mathcal {X}\) does cover \({\textbf{2}}^n_I\), then \(\mathcal {X}\diamondsuit {\textbf{2}}^n_I = \mathcal {X}\rtimes _\mu {\textbf{2}}^n_I\) (with \(\mu \) the mixing voltage) consists of two isomorphic copies of \(\mathcal {X}\), but the flags of these copies will be colored by the vertices of \({\textbf{2}}^n_I\). More precisely, if we colored the vertices of \({\textbf{2}}^n_I\) with white and black, then each vertex (flag) of \(\mathcal {X}\diamondsuit {\textbf{2}}^n_I\) is colored white or black according to its second coordinate. So \(\mathcal {X}\diamondsuit {\textbf{2}}^n_I\) consists of two copies of \(\mathcal {X}\) but together with an Icompatible coloring, and the two copies have opposite colorings. In particular, if X is orientable and \(I=\emptyset \), the two copies of \(\mathcal {X}\diamondsuit {\textbf{2}}^{n}_{\emptyset }\) are mirror images; if \(\mathcal {X}\) is a chiral maniplex, then \(\mathcal {X}\rtimes _\mu {\textbf{2}}^{n}_{\emptyset }\) consists of its two enantiomorphic forms.
Before ending this section let us note that the mixing voltages are the only voltages such that the product is naturally commutative. More precisely:
Proposition 3.6
Let \((\mathcal {X},\eta _\mathcal {X})\) and \((\mathcal {Y},\eta _\mathcal {Y})\) be (n, n)voltage operators. Then the function \((x,y)\mapsto (y,x)\) is an isomorphism between \(\mathcal {X}\rtimes _{\eta _{\mathcal {Y}}}\mathcal {Y}\) and \(\mathcal {Y}\rtimes _{\eta _{\mathcal {X}}}\mathcal {X}\) if and only if both \(\eta _\mathcal {X}\) and \(\eta _\mathcal {Y}\) are mixing voltages.
Proof
The alleged function is an isomorphism if and only if it maps \((x,y)^i = (\eta _\mathcal {Y}({}^{i}y)x,y^i)\) to \((y,x)^i = (\eta _\mathcal {X}({}^{i}x)y,x^i)\), for all \(x\in \mathcal {X}\), \(y\in \mathcal {Y}\) and \(i\in \{0,1,\ldots ,n1\}\). This means that \(\eta _\mathcal {Y}({}^{i}y) = r_i\) and \(r_i = \eta _\mathcal {X}({}^{i}x)\). \(\square \)
Note that \(\mathcal {X}\rtimes _{\eta _{\mathcal {Y}}}\mathcal {Y}\) and \(\mathcal {Y}\rtimes _{\eta _{\mathcal {X}}}\mathcal {X}\) may be isomorphic even if \(\eta _\mathcal {X}\) and \(\eta _\mathcal {Y}\) are not mixing voltages, for example, if \((\mathcal {X},\eta _\mathcal {X})\) is the orientable double cover and \((\mathcal {Y},\eta _\mathcal {Y})\) is the snub (or, of course, if \((\mathcal {X},\eta _\mathcal {X})=(\mathcal {Y},\eta _\mathcal {Y})\)).
3.2 Connectivity
We now turn our attention to the connectivity of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\). We say that an (n, m)voltage operator \((\mathcal {Y},\eta )\) preserves connectivity if whenever \(\mathcal {X}\) is connected \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is connected as well (in the context of [6], these are called fully stratified operations). We first analyze when we can find a path between two vertices of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) and use this to determine when a voltage operator preserves connectivity.
Lemma 3.7
Let (x, y) and \((x',y')\) be two vertices of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\). There is a path in \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) that starts at (x, y) and ends at \((x',y')\) if and only if there exists a path W in \(\mathcal {Y}\) that connects y with \(y'\) and such that \(\eta (W)x= x'\).
Proof
Assume there is a path \({}^{i_{k},\dots ,i_{1}} (x,y)\) that ends at \((x',y')\). By Remark 3.2 the path \({}^{i_{k}, \dots , i_{1}} (x,y)\) ends at \(\left( \eta (W)x, r_{i_{k}}\cdots r_{i_{1}}y\right) \), where W denotes the path \({}^{i_{k}, \dots , i_{1}} y\). Hence, \(\eta (W)x = x'\) and \(r_{i_{k}}\cdots r_{i_{1}}y = y'\). Moreover, \(r_{i_{k}}\cdots r_{i_{1}}y\) is precisely the endpoint of W, which implies that W connects y with \(y'\).
Conversely, assume that there exists a path W that connects y with \(y'\) and satisfies that \(\eta (W)x = x'\). Since W starts at y, it can be written as \(W={}^{i_{k}, \dots , i_{1}} y\), for some \(i_{k}, \dots , i_{1} \in \left\{ 0, \dots , m1 \right\} \). This implies that \(y' = r_{i_{k}}\cdots r_{i_{1}}y\). By Remark 3.2, the path \({}^{i_{k}, \dots , i_{1}} (x,y)\) ends at \(\left( \eta (W)x, r_{i_{k}}\cdots r_{i_{1}}y\right) = (x',y')\). \(\square \)
Proposition 3.8
The graph \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is connected if and only if \(\mathcal {Y}\) is connected and \(\eta (\Pi ^{y_0}(\mathcal {Y}))\) acts transitively on \(\mathcal {X}\) for some vertex \(y_0 \in \mathcal {Y}\).
Proof
We start by assuming that \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is connected, and let \(x,x'\in \mathcal {X}\). First note that \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) covers \(\mathcal {Y}\), so \(\mathcal {Y}\) must be connected as well. Now we want to prove that \(\eta (\Pi ^{y_0}(\mathcal {Y}))\) acts transitively on \(\mathcal {X}\). Since \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is connected, there is a path from \((x,y_0)\) to \((x',y_0)\). By Lemma 3.7 there exists \(W\in \Pi ^{y_0}(\mathcal {Y})\) such that \(\eta (W)x=x'\). Since x and \(x'\) were arbitrary, we have proved that \(\eta (\Pi ^{y_0}(\mathcal {Y}))\) acts transitively on \(\mathcal {X}\).
Now we assume that \(\mathcal {Y}\) is connected and \(\eta (\Pi ^{y_0}(\mathcal {Y}))\) acts transitively on \(\mathcal {X}\). We will prove that \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is connected. Let \(x_0 \in \mathcal {X}\) be fixed, and let \(x\in \mathcal {X}\) and \(y\in \mathcal {Y}\). Since \(\mathcal {Y}\) is connected, there is a path V from \(y_0\) to y in \(\mathcal {Y}\). Let \(\sigma =\eta (V)\). Since \(\eta (\Pi ^{y_0}(\mathcal {Y}))\) acts transitively on \(\mathcal {X}\), there exists \(W \in \Pi ^{y_0}(\mathcal {Y})\) such that \(\eta (W)x_0=\sigma ^{1}x\). Then the path WV goes from \(y_0\) to y and satisfies that \(\eta (WV)x_0=x\). By Lemma 3.7 the existence of the path WV implies that there exists a path in \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) from \((x_0,y_0)\) to (x, y). That is, \((x_0,y_0)\) is connected to every vertex of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\), implying that \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is connected. \(\square \)
Corollary 3.9
A voltage operator \((\mathcal {Y},\eta )\) preserves connectivity if and only if \(\mathcal {Y}\) is connected and \(\eta (\Pi ^{y_0}(\mathcal {Y})) = {{\,\textrm{Mon}\,}}(\mathcal {U})\) for some \(y_0\in \mathcal {Y}\).
In [4] the authors give a way to find generators of the automorphism group of a maniplex, given its symmetry type graph (STG). They do so they in terms of a spanning tree of the STG with trivial voltage on all its darts. Hence, it will be helpful to see Corollary 3.9 in the light of such a spanning tree of \(\mathcal {Y}\).
Corollary 3.10
Let \((\mathcal {Y},\eta )\) be a voltage operator. Assume that \(\mathcal {Y}\) is connected and has a spanning tree T such that all arcs in T have trivial voltage. Then \((\mathcal {Y},\eta )\) preserves connectivity if and only if \({{\,\textrm{Mon}\,}}(\mathcal {U})\) is generated by the voltages of the darts in \(\mathcal {Y}\) not in T.
Proof
The group \(\Pi ^{y_0}(\mathcal {Y})\) is generated by the paths \(C_d\) where d varies among darts in \(\mathcal {Y}\) not in T and \(C_d\) denotes the path that starts at \(y_0\), goes to the initial vertex of d through T, then uses the dart d, and then goes back to \(y_0\) through T. This implies that \(\eta (\Pi ^{y_0}(\mathcal {Y}))\) is generated by the voltages of those paths, but \(\eta (C_d)=\eta (d)\). \(\square \)
4 More Examples of Voltage Operations
Here, we shall see some examples of operations on maps and polytopes that can be seen as voltage operations. In fact, if we have an operation on polytopes (or maps) that can be seen as a voltage operation, by defining the corresponding voltage operator we have defined the operation on maniplexes and premaniplexes.
We show here that the Wythoffian constructions, including the snub, all prisms and pyramids over polytopes, the trapezotope and the kbubbles can be seen as voltage operations (each of them will be properly defined). Moreover, we also study the operations \(\mathcal {M}\mapsto 2^{\mathcal {M}}\) and \(\mathcal {M}\mapsto \hat{2}^{\mathcal {M}}\), and see that although there is no voltage operator that will give us these operations, whenever \(\mathcal {M}\) is regular, there is a way to describe the construction \(\hat{2}^{\mathcal {M}}\) as a voltage operation that depends on each \(\mathcal {M}\).
4.1 Wythoffian Constructions
We start by recalling the Wythoffian constructions from convex regular polytopes (see for example [2, 29]). Given a regular convex polytope, its symmetry group is generated by reflections \(\{\rho _0, \dots , \rho _{n1}\}\). We call these reflections the generating reflections of the group and note that their mirrors form a cone that, intersected with the polytope, results in a fundamental region for the polytope with respect to its symmetry group. For a Wythoffian construction, first choose a nonempty set A of generating reflections and pick a point v of the fundamental region that is not fixed by any of the reflections in A, but fixed by any generating reflection not in A (if they exist). The vertices of the new convex polytope are the orbit of v under \(\left\langle \rho _0, \dots \rho _{n1} \right\rangle \) and the polytope is the convex hull of such vertices. This can be represented on the Coxeter diagram (a graph with n nodes, each representing a generating reflection, and an edge labeled by the order of the product of their nodes, whenever it is bigger than 2) by marking the nodes representing the generating reflections in A.
If \(n=3\), then A is a nonempty subset of a 3 element set, so there are seven possibilities for A, each giving a different construction. In particular, for \(A = \{\rho _0\}\) and \(A = \{\rho _2\}\) the Wythoffian constructions give rise to the identity and dual operations, respectively, that have been given as voltage operations in Example 3.1. It is not difficult to generalize the Wythoffian constructions of regular polyhedra to maps on surfaces, by investigating the flag adjacencies of the constructed polyhedra. In particular, the medial and truncation of a map have been studied in [16, 17] and [24], respectively.
Figure 3 shows voltage operators \((\mathcal {Y}, \eta )\) for five of the Wythoffian constructions to be seen as a voltage operation. (As pointed out above, the two remaining ones correspond to when \(A = \{\rho _0\}\) and \(A = \{\rho _2\}\) that are the identity and dual operations, respectively.)
The omnitruncation in rank 3 (Fig. 3e) can be easily generalized to higher dimensions for a regular convex polytope \(\mathcal {P}\): it corresponds to the Wythoffian construction where A is the set of all reflections \(\{\rho _0, \rho _1, \dots , \rho _{n1}\}\). We can define this operation when \(\mathcal {P}\) is an abstract polytope (regular or not) in the following way. For \(i\in \{0,1,\ldots ,n1\}\), the ifaces of \({{\,\mathrm{{ot}}\,}}(\mathcal {P})\) are chains of \(\mathcal {P}\) that do not contain the greatest nor least elements and have \(ni\) elements, and given two faces C and \(C'\) of \({{\,\mathrm{{ot}}\,}}(\mathcal {P})\) we say that \(C\le _{{{\,\mathrm{{ot}}\,}}(\mathcal {P})} C'\) if and only if \(C'\subset C\). In particular, the vertices of \({{\,\mathrm{{ot}}\,}}(\mathcal {P})\) are the flags of \(\mathcal {P}\) and an edge of the form \(\Phi \setminus \{\Phi _i\}\) joins the vertices \(\Phi \) and \(\Phi ^i\). This means that the 1skeleton of \({{\,\mathrm{{ot}}\,}}(\mathcal {P})\) is the flag graph of \(\mathcal {P}\). Another way to construct \({{\,\mathrm{{ot}}\,}}(\mathcal {P})\) is as the colorful polytope of the flag graph of \(\mathcal {P}\) [1].
The omnitruncation in rank n can be thought as a voltage operation. To see this, use the following construction for the voltage operator. Let \(\mathcal {S}\) be the flag graph of an \((n1)\)simplex. First, change the color of each edge by increasing it by 1 (i.e. if two flags of \(\mathcal {S}\) are iadjacent, they will be joined by an edge of color \(i+1\)). All these edges will have the identity voltage. Note that now each vertex has an edge of color i, for \(i\in \{1,2,\dots , n1\}\). Now, label each connected component of the graph with edges of colors \( 1, \dots , n2\) with a different number from 0 to \(n2\). Then, add a semiedge of color 0 at each vertex with voltage \(r_j\) where j is the label of connected component of the vertex. The resulting voltage operator \((\mathcal {Y},\eta )\) satisfies that the flag graph of \({{\,\mathrm{{ot}}\,}}(\mathcal {P})\) is \(\mathcal {P}\rtimes _\eta \mathcal {Y}\).
As can be expected all Wythoffian constructions for higher dimensions can be seen as voltage operations. We do not give here all the voltage operators, as for rank n, there can be up to n! vertices in it. However, in Sect. 7 we describe a theoretical way of finding them all.
The snub of a polyhedron can be thought as a generalized Wythoffian construction, where the rotational subgroup is used. We shall not give details of this kind of constructions, but rather note some interesting differences between the snub and the Wythoffian described above.
The snub operator given in Fig. 4. In contrast with the Wythoffian operations that preserve connectivity of any maniplex, the snub operator only preserves connectivity for nonorientable premaniplexes. If we apply this snub operation to an orientable premaniplex \(\mathcal {X}\) (in particular to a regular convex polyhedron) we get two copies of what is usually referred to as the snub \(\mathcal {X}\), each copy having all the “rotational” symmetry of \(\mathcal {X}\) but not the reflection symmetry of \(\mathcal {X}\). We will explore this phenomenon deeper in Sect. 5.
4.2 Prisms and Pyramids Over Polytopes
Let \(\mathcal {P}\) be an abstract polytope. The pyramid over \(\mathcal {P}\), denoted by \({{\,\textrm{Pyr}\,}}(\mathcal {P})\) is the poset \(\mathcal {P}\times \{0,1\}\) with the product order, that is, \((F,\ell )\le (G,\ell ')\) if and only if \(F\le G\) and \(\ell \le \ell '\). Similarly, the prism over \(\mathcal {P}\), denoted by \({{\,\textrm{Pri}\,}}(\mathcal {P})\) is defined as the poset \(((\mathcal {P}\setminus \mathcal {P}_{1})\times \Lambda ) \cup \{F_{1}\}\) where \(\Lambda \) is the poset \(\{\{\lambda _0\},\{\lambda _1\},\{\lambda _0,\lambda _1\}\}\) ordered by inclusion and \(F_{1}\) is the unique minimum element of the prism.
Both the prism and the pyramid over an npolytope are \((n+1)\)polytopes. They are particular cases of products of polytopes, in the sense of [10]: the pyramid is the join product by a vertex, while the prism is the direct product by an edge.
One can see that each flag \(\Psi \) in \({{\,\textrm{Pyr}\,}}(\mathcal {P})\) is of the form
where \(\Phi \) is a flag in \(\mathcal {P}\) and \(t\in \{1,0,1,\ldots ,n\}\). Hence, we identify the set of flags of \({{\,\textrm{Pyr}\,}}(\mathcal {P})\) with \(\mathcal {F}(\mathcal {P})\times \{1,0,\ldots ,n\}\).
In a similar way, one can identify the set of flags of \({{\,\textrm{Pri}\,}}(\mathcal {P})\) with \(\mathcal {F}\times \{0,1,\ldots ,n\}\times \{\lambda _0,\lambda _1\}\), where the last coordinate tells us if the 0face is of the form \((\Phi _0,\{\lambda _0\})\) or \((\Phi _0,\{\lambda _1\})\).
Let us first observe the pyramid. The iadjacencies of the flags (see [10]), for \(i \in \{0,1,\dots , n\}\), are given by
Let \((\mathcal {Y},\eta )\) be the \((n,n+1)\)voltage operator whose vertices are the numbers \(\{1,0,\ldots ,n\}\) with the iadjacencies (for \(i \in \{0,1,\dots , n\}\)) given by:
and the voltage assignment given by:
(see Fig. 5). Then the flag graph of \({{\,\textrm{Pyr}\,}}(\mathcal {P})\) is \(\mathcal {P}\rtimes _\eta \mathcal {Y}\).
We now turn our attention to the prism, where the iadjacencies of the flags (see [10]), for \(i \in \{0,1,\dots , n\}\), are given by
where \(\lambda '\) is \(\lambda _1\) if \(\lambda =\lambda _0\) and vice versa.
Let \((\mathcal {Y}',\eta ')\) be the \((n,n+1)\)voltage operator whose vertices are \(\{0,\ldots ,n\}\times \{\lambda _0,\lambda _1\}\) with the adjacencies given by:
and the voltage assignment given by:
We show this operator in Fig. 6.
The flag graph of \({{\,\textrm{Pri}\,}}(\mathcal {P})\) is \( \mathcal {P}\rtimes _{\eta '}\mathcal {Y}'\).
4.2.1 The Trapezotope
Given a polytope \(\mathcal {P}\), we define the trapezotope over \(\mathcal {P}\) denoted by \({{\,\mathrm{{Trp}}\,}}(\mathcal {P})\) as follows: the faces of rank i, for \(i \ge 0\), of the trapezotope are the ordered pairs (F, G) where F and G are faces of \(\mathcal {P}\), \(F\le G\) and \({{\,\textrm{rk}\,}}(G){{\,\textrm{rk}\,}}(F)=i\). The order in \({{\,\mathrm{{Trp}}\,}}(\mathcal {P})\) is defined as \((F,G) \le (F',G')\) if and only if \(F' \le F\) and \(G \le G'\). Those faces together with an additional formal face of rank \(1\) define a polytope.
The vertices of \({{\,\mathrm{{Trp}}\,}}(\mathcal {P})\) are the pairs (F, F), so they are in correspondence to the faces of \(\mathcal {P}\), and the edges of \({{\,\mathrm{{Trp}}\,}}(\mathcal {P})\) are in correspondence to the edges of the Hasse diagram of \(\mathcal {P}\). Thus, the 1skeleton of the trapezotope of \(\mathcal {P}\) is the Hasse diagram of \(\mathcal {P}\). The trapezotope over \(\mathcal {P}\) is in fact the dual of the antiprism over \(\mathcal {P}\), defined in [11].
In [11] the authors showed that the trapezotope of a polytope \(\mathcal {P}\) (or rather its dual) is in fact a polytope and that the flags are in onetoone correspondence with the set \(\mathcal {F}(\mathcal {P})\times \mathbb {Z}_2^{n+1}\). When we denote a flag of \({{\,\mathrm{{Trp}}\,}}(\mathcal {P})\) by \((\Phi ,v)\), the ith coordinate of the vector v tells us if the faces \((\Phi ,v)_i=(F_i,G_i)\) and \((\Phi ,v)_{i1}=(F_{i1},G_{i1})\) differ in the first coordinate (when \(v_i=0\)) or in the second one (when \(v_i=1\)). Using this natural correspondence, one can prove that the iadjacent flag of the flag \((\Phi ,v)\) is:
where j is the rank of \(F_i\) if \(v_i=0\) or the rank of \(G_i\) if \(v_i=1\) and \(e_{i}\) denotes the vector in \(\mathbb {Z}_2^{n+1}\) whose all but the ith coordinate are zero. Let \(\mathcal {Y}\) be the premaniplex whose vertices are \(\mathbb {Z}_2^n\) and the adjacencies are given by:
And let \(\eta \) be the voltage assignment given by
where j is defined the same way as before. One can confirm that the voltage operator \((\mathcal {Y},\eta )\) satisfies that for every polytope \(\mathcal {P}\), the flag graph of \({{\,\mathrm{{Trp}}\,}}(\mathcal {P})\) is \( \mathcal {P}\rtimes _\eta \mathcal {Y}\).
4.2.2 The kBubble
The kbubble operation was introduced by Helfand in [13] as a generalization of the truncation of the vertices.
Let \(\mathcal {P}\) be an abstract npolytope. The kbubble \([\mathcal {P}]_k\) of \(\mathcal {P}\) is defined as follows. Let \(\mathcal {P}_i\) denote the set of ifaces of \(\mathcal {P}\). The set \(([\mathcal {P}]_k)_i\) of ifaces of \([\mathcal {P}]_{k}\) is defined by
The order in \([\mathcal {P}]_k\) is defined by the following rules: Let H and \(H'\) be faces in \(\mathcal {P}\) with ranks different than k, let F be a kface of \(\mathcal {P}\) and let G and \(G'\) be faces of \(\mathcal {P}\) properly containing F.

\(H<H'\) in \([\mathcal {P}]_k\) if and only if \(H<H'\) in \(\mathcal {P}\).

\(H<(F,G)\) in \([\mathcal {P}]_k\) if and only if \(H<F\) in \(\mathcal {P}\).

\((F,G)<H\) in \([\mathcal {P}]_k\) if and only if \(G\le H\) in \(\mathcal {P}\).

\((F,G)<(F,G')\) in \([\mathcal {P}]_k\) if and only if \(G<G'\) in \(\mathcal {P}\).
One can see that the flags of \([\mathcal {P}]_k\) are of the form
where \(\{F_0,\ldots ,F_{n1}\}\) is a flag in \(\mathcal {P}\). So every flag of \([\mathcal {P}]_k\) is determined by a flag \(\Phi \) of \(\mathcal {P}\) and a number \(\ell \in \{k+1,\ldots ,n\}\). If we denote this flag by \((\Phi ,\ell )\) then the flag adjacencies on \([\mathcal {P}]_k\) are described by the following rules:
With this information we can see that the kbubble is described by the voltage operator in Fig. 8.
4.2.3 The \(2^\mathcal {M}\) and \(\hat{2}^\mathcal {M}\) Operations
Given a finite maniplex \(\mathcal {M}\) of rank n, the maniplex \(2^\mathcal {M}\) was defined in [8] as follows: Label the 0faces of \(\mathcal {M}\) with the set \(\{1, \dots , \ell \}\). The flag set of \(2^\mathcal {M}\) is \(\mathcal {M}\times \mathbb {Z}_2^{\ell }\). Then adjacencies of a flag \((\Psi , v)\), with \(\Psi \in \mathcal {M}\) and \(v \in \mathbb {Z}_2^{\ell }\), are given by the following formula, where \(e_{k} \in \mathbb {Z}_2^{\ell }\) is the vector that has 0 in all its entries except the kth one:
It is wellknown that if \(\mathcal {M}\) is the flag graph of a polytope \(\mathcal {P}\), then \(2^\mathcal {M}\) is the flag graph of the polytope \(2^\mathcal {P}\) discovered by Danzer [7] and first defined in [28] (see [20, sect. 8C] and [22] for details).
The maniplex \(\hat{2}^\mathcal {M}\) is defined as the dual of \(2^{\mathcal {M}^{*}}\), where \(\mathcal {M}^{*}\) is the dual of \(\mathcal {M}\). Thus, the set of flags of \(\hat{2}^\mathcal {M}\) is \(\mathcal {M}\times \mathbb {Z}_2^{m}\), where the set of facets of \(\mathcal {M}\) is labeled with the set \(\{1, \dots , m\}\). The adjacencies in \(\hat{2}^\mathcal {M}\) are given by:
where \(e_{k}\in \mathbb {Z}_2^{m}\) is the vector that has 0 in all its entries except the kth one.
These operations on maniplexes are very interesting. In particular in the context of voltage operations, Example 5.3 will show that one cannot find a premaniplex \(\mathcal {Y}\) that satisfies that \(2^\mathcal {M}= \mathcal {M}\rtimes _\eta \mathcal {Y}\), for every maniplex \(\mathcal {M}\) (not even if we fix the rank of \(\mathcal {M}\), or ask \(\mathcal {M}\) to be regular). However we shall see that if \(\mathcal {M}\) is a regular premaniplex, then there exists a premaniplex \(\mathcal {Y}_\mathcal {M}\) and a voltage assignment \(\eta \) such that \(\hat{2}^\mathcal {M}= \mathcal {M}\rtimes _\eta \mathcal {Y}_\mathcal {M}\).
Let \(\mathcal {M}\) be a regular npremaniplex and let \(\rho _0, \rho _1, \dots , \rho _{n1}\) be the distinguished generators of \(\Gamma (\mathcal {M})\) with respect to the base flag \(\Phi \). We define the premaniplex \(\mathcal {Y}:=\mathcal {Y}_\mathcal {M}\) as follows. The set of vertices of \(\mathcal {Y}\) is the set \(\mathbb {Z}_2^{m}\). Given \(v \in \mathcal {Y}\) the entries of v correspond to facets of \(\mathcal {M}\). Assume that the facets of \(\mathcal {M}\) are labeled with the set \(\{1, \dots , m\}\), as above, in such a way that the facet corresponding to the base flag is labeled with 1. Notice that every element in \(\Gamma (\mathcal {M})\) permutes the facets of \(\mathcal {M}\), which in turn induces an action of \(\Gamma (\mathcal {M})\) on \(\mathbb {Z}_2^{m}\) by permutation of coordinates. More precisely, for \(\alpha \in \Gamma (\mathcal {M})\) and \(v=(v_1, \dots v_{m}) \in \mathbb {Z}_2^{m}\), let \(v \alpha \) denote the vector \((v_{1\alpha ^{1}}, \dots , v_{m\alpha ^{1}})\), that is, the vector resulting from v after permuting the coordinates according to \(\alpha \).
The adjacencies in \(\mathcal {Y}\) are given as follows: for \(i\in \{0,1,\dots , n1\}\) there is an edge of color i between v and \(v^i:=v \rho _{i}\). Further, there is an edge of color n between v and \(v+e_{1}\). We now define the voltage assignment on \(\mathcal {Y}\) as:
We shall show that \(\hat{2}^\mathcal {M}\cong \mathcal {M}\rtimes _\eta \mathcal {Y}\). For this, we recall that \(\mathcal {M}\) is regular, and hence any flag of \(\mathcal {M}\) can be written in a unique way as \(\Phi \alpha \), where \(\Phi \) is the base flag and \(\alpha \in \Gamma (\mathcal {M})\).
Consider \(\varphi :\hat{2}^\mathcal {M}\rightarrow \mathcal {M}\rtimes _\eta \mathcal {Y}\) defined as
Then, \(\varphi \) preserves incidences. In fact, we observe that if \(k= 1\alpha \):
On the other hand, for \(i<n\) we have that:
The computation above shows that \(\hat{2}^\mathcal {M}\) and \(\mathcal {M}\rtimes _\eta \mathcal {Y}\) are isomorphic.
It is clear that if we drop the regularity condition on \(\mathcal {M}\) the given voltage operator \((\mathcal {Y}, \eta )\) is not welldefined as it depends heavily on the existence of the distinguished generators of the automorphism group. Thus, we can ask ourselves the following questions.
Question 4.1
Is it true that for every maniplex \(\mathcal {M}\), there exists a voltage operator \((\mathcal {Y}_\mathcal {M},\eta )\) such that \(\hat{2}^\mathcal {M}\cong \mathcal {M}\rtimes _\eta \mathcal {Y}_\mathcal {M}\)?
For example, we have not been able to determine if there exists such voltage operator for the case when \(\mathcal {M}\) is the pyramid over a digon. If the answer for the above question is negative, then we could ask:
Question 4.2
Give necessary and sufficient conditions on a maniplex \(\mathcal {M}\) in such a way that there exists a voltage operator \((\mathcal {Y}_\mathcal {M},\eta )\) such that \(\hat{2}^\mathcal {M}\cong \mathcal {M}\rtimes _\eta \mathcal {Y}_\mathcal {M}\).
Note that a sufficient condition for the last question is to have \(\mathcal {M}\) to be regular, as shown above. However chances are this condition is not necessary. For example, we believe that if the symmetry type graph of \(\mathcal {M}\) is itself a regular premaniplex, the above voltage operator might exist.
5 Automorphisms
In this section we will study the interplay between the automorphism group of a premaniplex and the resulting premaniplex after a voltage operation, as well as their symmetry type graphs.
Let \(\mathcal {X}\) be a premaniplex and let \((\mathcal {Y}, \eta )\) be a voltage operator. Observe that \(\Gamma (\mathcal {X})\) acts faithfully as a group of automorphisms on \(\mathcal {X}\rtimes _\eta \mathcal {Y}\). Indeed, if \(\gamma \in \Gamma (\mathcal {X})\), then the mapping \({\bar{\gamma }}: (x,y) \mapsto (x\gamma , y)\) induces an automorphism of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\). To see this, just note that \({\bar{\gamma }}\) commutes with the monodromies
Thus, \(\Gamma (\mathcal {X}) \le \Gamma (\mathcal {X}\rtimes _\eta \mathcal {Y})\) for every voltage operator \((\mathcal {Y}, \eta )\) and we may replace \({\bar{\gamma }}\) by \(\gamma \) to simplify notation. The latter result implies that \(\Gamma (\mathcal {X})\) acts by automorphisms on \(\mathcal {X}\rtimes _\eta \mathcal {Y}\). Note that \(\mathcal {X}\) naturally covers \(\mathcal {X}/\Gamma \) for every group \(\Gamma \le \Gamma (\mathcal {X})\). As expected \(\mathcal {X}\rtimes _\eta Y\) naturally covers \((\mathcal {X}\rtimes _\eta Y)/\Gamma \), In fact, we have a slightly more general result:
Theorem 5.1
Let \(\mathcal {X}\) and \(\mathcal {X}'\) be npremaniplexes, and let \((\mathcal {Y},\eta )\) a voltage operator. If \(\mathcal {X}\) covers \(\mathcal {X}'\), then \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) covers \(\mathcal {X}'\rtimes _\eta \mathcal {Y}\).
In particular, when \(\mathcal {X}'\) is obtained from \(\mathcal {X}\) as a quotient by a group \(\Gamma \le \Gamma (\mathcal {X})\), we get that
Proof
Let \(\varphi :\mathcal {X}\rightarrow \mathcal {X}'\) be a covering homomorphism. If \(\mathcal {X}'=\mathcal {X}/\Gamma \) we chose \(\varphi \) as the natural projection \(x\mapsto x\Gamma \). Define \(\varphi ': \mathcal {X}\rtimes _\eta \mathcal {Y}\rightarrow \mathcal {X}'\rtimes _\eta \mathcal {Y}\) by \(\varphi ': (x,y) \mapsto (\varphi (x), y)\).
Now, using the fact that \(\varphi \) is a homomorphism we simply check that
This proves that \(\varphi '\) is a homomorphism, and it is trivial to see that it is surjective. Therefore it is a covering.
In the particular case when \(\mathcal {X}'=\mathcal {X}/\Gamma \) we get that \(\varphi '(x,y) = (x\Gamma ,y)\), but by using the natural action of \(\Gamma \) on \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) we get that \((x\Gamma ,y) = (x,y)\Gamma \). \(\square \)
The above theorem is of particular interest when \(\mathcal {X}\) is a maniplex and \(\Gamma =\Gamma (\mathcal {X})\). In this case, \(\mathcal {T}:=\mathcal {X}/\Gamma \) is the symmetry type graph of \(\mathcal {X}\). Theorem 5.1 tells us that the symmetry type graph of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) with respect to \(\Gamma \) is precisely \(\mathcal {T}\rtimes _\eta \mathcal {Y}\). However it is important to remark that \(\Gamma \) might be a proper subgroup of \(\Gamma (\mathcal {X}\rtimes _\eta \mathcal {Y})\), that is, \(\mathcal {T}\rtimes _\eta \mathcal {Y}\) might not be the symmetry type graph of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\), even when \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is a maniplex. An example of such situation will be given in Example 5.5. This phenomenon will be explored deeply in [15].
Words of caution: the fact that \(\Gamma (\mathcal {X})\) acts by automorphisms on \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) does not imply that it acts by automorphisms on each connected component of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\). For example, as pointed out in Sect. 4, when \(\mathcal {X}\) is an orientable (pre)maniplex of rank 3 and \((\mathcal {Y},\eta )\) is the snub operator of Fig. 4, \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) has two connected components, say the left snub (whose flags are those of the form (x, y) with \(x\in \mathcal {X}\) a white flag), and the right snub (whose flags are those of the form (x, y) with \(x\in \mathcal {X}\) a black flag). Each component has all the “rotational” symmetry of \(\mathcal {X}\) but none of the “reflection” symmetries of \(\mathcal {X}\). However, if there exists a reflection \(\tau \in \Gamma (\mathcal {X})\), that is, an automorphism that interchanges white flags with black flags, then \(\tau \) acts on \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) by swapping the left snub with the right snub. This means that if we consider rooted premaniplexes \((\mathcal {X},x)\) and \((\mathcal {Y},y)\), then the rooted premaniplex \((\mathcal {X}\rtimes _\eta \mathcal {Y}, (x,y))\) may not have all the symmetries of \(\mathcal {X}\). In fact we have the following result:
Proposition 5.2
Let \((\mathcal {X},x)\) be a rooted premaniplex, \(((\mathcal {Y},y),\eta )\) a rooted voltage operator and \(\gamma \in \Gamma (\mathcal {X})\). Then \(\gamma \in \Gamma (\mathcal {X}\rtimes _\eta \mathcal {Y}, (x,y))\) if and only if there exists a closed path W based on y such that \(x\gamma = \eta (W)x\).
Proof
Assume that \(\gamma \in \Gamma (\mathcal {X}\rtimes _\eta \mathcal {Y}, (x,y))\). This means that \((x,y)\gamma \) is an element of \((\mathcal {X}\rtimes _\eta \mathcal {Y}, (x,y))\); in other words, \((x,y)\gamma \) and (x, y) are in the same connected component of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\). Hence, there exists a monodromy w such that \(w(x,y) = (x,y)\gamma \). Note that on one hand, \((x,y)\gamma = (x\gamma , y)\), while on the other hand \(w(x,y) = (\eta ({}^{w} y)x, w y)\). Thus, \((x\gamma , y) = (\eta ({}^{w} y)x, w y)\), which implies that \(y=w y\) and therefore \(W:= {}^{w} y \in \Pi ^y(\mathcal {Y})\).
Conversely, suppose that there exists \(W\in \Pi ^y(\mathcal {Y})\) such that \(x\gamma = \eta (W)x\). The fact that \(W\in \Pi ^y(\mathcal {Y})\) implies that there exists a monodromy w such that \(w y = y\) and \(W={}^{w} y\). Hence, \((x,y)\gamma = (x\gamma , y) = (\eta (W)x, y) = (\eta ({}^{w} y)x, y) = (\eta ({}^{w} y)x, w y) = w(x,y)\), which implies that \((x,y)\gamma \) and (x, y) are in the same connected component of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\). Since we already know that \(\gamma \) is an automorphism of \(\mathcal {X}\rtimes _\eta \mathcal {Y}\), this implies that \(\gamma \in \Gamma (\mathcal {X}\rtimes _\eta \mathcal {Y}, (x,y))\). \(\square \)
Note that we may use Theorem 5.1 to determine when an operation is not a voltage operation, as we see in the following example.
Example 5.3
In this example we will prove that there is no \((n,n+1)\)voltage operator \((\mathcal {Y},\eta )\) such that \(2^\mathcal {M}= \mathcal {M}\rtimes _\eta \mathcal {Y}\) for every nmaniplex \(\mathcal {M}\) (as was hinted in Question 4.1). For a 2gon \(\{2\}\), the polytope \(2^{\{2\}}\) is the quadrangular dihedron \(\{4,2\}\) (the map on the sphere consisting of two squared faces sharing all the vertices and edges). For the square \(\{4\}\), we have that \(2^{\{4\}}\) is the toroidal map \(\{4,4\}_{(4,0)}\) (a \(4\times 4\) grid on the torus). We can get a 2gon from a square by taking its quotient by the cyclic group of order two generated by the halfturn around the center, that is \(\{2\}=\{4\}/\mathbb {Z}_2\). However, it is not possible to get the quadrangular dihedron by taking a quotient of \(\{4,4\}_{(4,0)}\) by any group of order 2. In other words \(2^{\{4\}} / \mathbb {Z}_2 \ncong 2^{\{4\}/\mathbb {Z}_2}\). Hence, Theorem 5.1 is not satisfied for the operation \(2^\mathcal {M}\). Thus, there is no voltage operator \((\mathcal {Y},\eta )\) such that \(2^\mathcal {M}= \mathcal {M}\rtimes _\eta \mathcal {Y}\) for every maniplex \(\mathcal {M}\).
The previous example uses Theorem 5.1 to prove that there does not exist a voltage operation \((\mathcal {Y}, \eta )\) such that \(2^\mathcal {M}\cong \mathcal {M}\rtimes _\eta \mathcal {Y}\) for all maniplexes \(\mathcal {M}\), in other words, that \(\mathcal {M}\mapsto 2^\mathcal {M}\) is not a voltage operation. Observe that this fact can also be established in a more elementary way by simply comparing the number of flags in \(\mathcal {M}\rtimes _\eta \mathcal {Y}\) and in \(2^{\mathcal {M}}\). We use Theorem 5.1 to emphasize the relation between quotients and voltage operations. In fact, Theorem 5.1 characterizes all voltage operations; this shall be shown in Theorem 7.3.
It is easy to come up with examples of operations that are not voltage operations, for instance, by defining them differently for different cases. However, the operation \(\mathcal {M}\mapsto 2^\mathcal {M}\) is of interest because it is a functor between the categories \(\textrm{pMpx}^n\) and \(\textrm{pMpx}^{n+1}\), in other words, if there is a homomorphism \(p:\mathcal {M}\rightarrow \mathcal {M}'\) then there is a natural way to define a homomorphism \({\tilde{p}}:2^\mathcal {M}\rightarrow 2^{\mathcal {M}'}\), as \({\tilde{p}}(\Phi ,v)\mapsto (p(\Phi ),u)\) where
It is straightforward prove that \({\tilde{p}}\) is a maniplex homomorphism and that \(p\mapsto {\tilde{p}}\) is indeed a functor.
Let \(\mathcal {M}\) be a premaniplex, let \(\Gamma \) be a group of automorphisms of \(\mathcal {M}\) and let \(\mathcal {X}=\mathcal {M}/\Gamma \). Hence, there exists a voltage assignment \(\xi :\Pi (\mathcal {X})\rightarrow \Gamma \) such that \(\mathcal {X}^\xi \) is isomorphic to \(\mathcal {M}\), where \(\mathcal {X}^\xi \) is the derived graph (see Sect. 2.1). If \((\mathcal {Y},\eta )\) is a voltage operation, Theorem 5.1 tells us that \((\mathcal {M}\rtimes _\eta \mathcal {Y})/\Gamma \) is isomorphic to \(\mathcal {X}\rtimes _\eta \mathcal {Y}\). Then, there exists a voltage assignment \(\theta :\Pi (\mathcal {X}\rtimes _\eta \mathcal {Y})\rightarrow \Gamma \) such that \((\mathcal {X}\rtimes _\eta \mathcal {Y})^\theta \) is isomorphic to \(\mathcal {M}\rtimes _\eta \mathcal {Y}\) (again, \((\mathcal {X}\rtimes _\eta \mathcal {Y})^\theta \) denotes the derived graph). The following theorem tells us how to find \(\theta \) in terms of \(\eta \) and \(\xi \).
Theorem 5.4
Let \((\mathcal {X},\xi )\) be a voltage premaniplex with voltage group \(\Gamma \) and let \((\mathcal {Y},\eta )\) be a voltage operator.
Define \(\theta = \theta (\eta ,\xi ):\Pi (\mathcal {X}\rtimes _\eta \mathcal {Y})\rightarrow \Gamma \) as follows:
for every \(w \in {{\,\textrm{Mon}\,}}(\mathcal {U})\).
Then \((\mathcal {X}\rtimes _\eta \mathcal {Y})^\theta \) is isomorphic to \(\mathcal {X}^\xi \rtimes _\eta \mathcal {Y}\).
Proof
Start by noticing that, by definition, \(r_i(x,\gamma ) = (r_ix, \xi (^ix)\gamma )\) and similarly \(r_i((x,y),\gamma ) = (r_i(x,y), \theta (^i(x,y))\gamma )\). Now, define the function
given by
We shall show that \(\varphi \) is a premaniplex isomorphism.
Since both \(\mathcal {X}^\xi \) and \((\mathcal {X}\rtimes _\eta \mathcal {Y})^\theta \) have \(\Gamma \) as voltage group, \(\varphi \) is a bijection. We should see next that \(\varphi \) preserves iadjacencies for \(i\in \{0,1,\ldots ,m1\}\). In fact:
\(\square \)
Example 5.5
It is not difficult to see that the antiprism of a qgon can be obtained by taking the medial of the pyramid over a qgon. Thus, we shall use Theorem 5.4 to recover the antiprism of a qgon as a voltage maniplex. To do so, we first construct the pyramid over a qgon as the derived graph of a voltage premaniplex and then apply the medial operator (as a voltage operator).
Note that the automorphism group of a qgonal pyramid coincides with the automorphism group of its base, which is the dihedral group
where \(\rho _0\) is thought as the reflection in a plane orthogonal to the base, that includes the midpoint of an edge e in the base; and \(\rho _1\) as the reflection in a plane orthogonal to the base, that includes one of the vertices incident to e. Hence, the qgonal pyramid can be recovered from its symmetry type graph \(\mathcal {X}\) via the voltage premaniplex \((\mathcal {X},\xi )\) shown in Fig. 9. (As in previous examples, the colors red, green and blue for the edges represent 0, 1 and 2adjacencies, respectively.)
Let \((\mathcal {Y},\eta )\) be the medial operator. Then, by Theorem 5.4, \((\mathcal {X}\rtimes _\eta \mathcal {Y},\theta )\) is the antiprism over a qgon, where \(\theta \) is defined as in the theorem.
Theorem 5.4 gives us a way to construct such antiprism. When we want to calculate, for example, the endpoint and voltage of the dart \({}^{0} (a,x)\) (the red dart that starts at (a, x)), we first see that \(\eta ({}^{0} x) = r_1\), so we follow the path \({}^{1} a\) and we see that it ends in a and has voltage \(\xi ({}^{1} a) = \rho _1\). Therefore, \((a,x)^0 = (\eta ({}^{0} x)a,x^0) = (a,x)\) (i.e., the dart is a semiedge) and the dart \({}^{0} (a,x)\) has voltage \(\theta ({}^{0} (a,x)) = \rho _1\).
Note that \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is the symmetry type graph of the antiprism with respect to the automorphism group of its base, but not with respect to the full automorphism group of the antiprism. In fact, the antiprism over any polygon always has extra symmetry, which is induced by the isomorphism between the base of the antiprism and its dual, so antiprisms over polygons have usually four orbits on flags. The antiprism over the triangle is in fact an octahedron and therefore regular, so in that case the actual symmetry type graph is \({\textbf{1}}^3\).
A consequence of Theorem 5.4 is the following corollary:
Corollary 5.6
Let \(\mathcal {X}\) be a regular npremaniplex with automorphism group \(\langle \rho _{0}, \dots , \rho _{n1} \rangle \) and let \((\mathcal {Y},\eta )\) be a (n, m)voltage operator. Then \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is isomorphic to the derived graph \(\mathcal {Y}^\nu \), where \(\nu :\Pi (\mathcal {Y})\rightarrow \Gamma (\mathcal {X})\) is the voltage assignment obtained from \(\eta \) by replacing each \(r_i\) with \(\rho _i\).
Proof
Since \(\mathcal {X}\) is regular, it is isomorphic to the derived graph \(({\textbf{1}}^n)^\xi \) where, if x is the only vertex of \({\textbf{1}}^n\), \(\xi ({}^{i}x)=\rho _i\). Then Theorem 5.4 tells us that \(\mathcal {X}\rtimes _\eta \mathcal {Y}\) is isomorphic to \(({\textbf{1}}^n\rtimes _\eta \mathcal {Y})^\theta \) with \(\theta ({}^{w} (x,y)) = \xi ({}^{{\eta ({}^{w} y}} ){x})\). This means precisely that \(\theta \) replaces each occurrence of \(r_i\) in \(\eta \) by the voltage of the semiedge of color i in \({\textbf{1}}^n\), but this is exactly \(\rho _i\). By applying the natural isomorphism \({\textbf{1}}^n \rtimes _\eta \mathcal {Y}\rightarrow \mathcal {Y}\) we get the desired result. \(\square \)
6 Composition of Voltage Operations
The result of applying a voltage operation to a premaniplex is again a premaniplex. Thus, it is natural to think about the composition of two voltage operations. In contrast, the result of applying a voltage operation to a maniplex is not always a maniplex, so one must be careful with this fact when composing operations, as the result of a voltage operation can be disconnected.
It is interesting to note that in fact the composition of two voltage operations can be written as a new voltage operation. In this section we describe how to do this by using the operator \(\theta \) defined in Theorem 5.4, but instead of using an arbitrary voltage premaniplex \((\mathcal {X},\xi )\) and a voltage operator \((\mathcal {Y},\eta )\) we use two voltage operators \((\mathcal {Y}_1,\eta _1)\) and \((\mathcal {Y}_2,\eta _2)\).
Theorem 6.1
Let \(\mathcal {X}\) be an npremaniplex, \((\mathcal {Y}_1,\eta _1)\) an (n, m)voltage operator and \((\mathcal {Y}_2,\eta _2)\) a \((m,\ell )\)voltage operator. Then
where \(\theta =\theta (\eta _2,\eta _1)\) is defined as in Equation (5.1).
Proof
We define \(\varphi :(\mathcal {X}\rtimes _{\eta _1} \mathcal {Y}_1)\rtimes _{\eta _2} \mathcal {Y}_2 \rightarrow \mathcal {X}\rtimes _{\theta } (\mathcal {Y}_1\rtimes _{\eta _2} \mathcal {Y}_2)\) in the natural way, that is,
It is clear that \(\varphi \) is a bijection, so we only need to prove that it commutes with \(r_i\), for all \(i\in \{0,1,\ldots ,\ell 1\}\):
\(\square \)
The above theorem can be applied in different contexts, we give some examples here. Let \(({\textbf{1}}^m,d)\) be the duality operator and consider a (n, m)voltage operator \((\mathcal {Y},\eta )\). If \(\mathcal {X}\) is a premaniplex, then \((\mathcal {X}\rtimes _\eta {\mathcal {Y}})\rtimes _{d} {\textbf{1}}^m\) is the dual of \((\mathcal {X}\rtimes _\eta {\mathcal {Y}})\). Theorem 6.1 tells us that this dual is in fact isomorphic to \(\mathcal {X}\rtimes _{\theta }(\mathcal {Y}\rtimes _{d}{\textbf{1}}^m)\). Thus, \((\mathcal {Y}\rtimes _{d}{\textbf{1}}^m,\theta )\) is the dual of \(\mathcal {Y}\), where the voltages of the darts are the same as the ones of the darts of the dual color in \((\mathcal {Y},\eta )\). In other words, if \((\mathcal {Y}^*,\eta ^*)\) denotes the voltage operator we get by recoloring the darts of color i in \(\mathcal {Y}\) with the color \(n1i\), then the dual of \(\mathcal {X}\rtimes _{\mathcal {Y}}\) is \(\mathcal {X}\rtimes _{\eta ^*}\mathcal {Y}^*\).
More generally, Theorem 6.1 lets us define the composition of two operators. If \((\mathcal {Y}_1,\eta _1)\) is an (n, m)operator and \((\mathcal {Y}_2,\eta _2)\) is an \((m,\ell )\)operator we define the composition of \((\mathcal {Y}_1,\eta _1)\) with \((\mathcal {Y}_2,\eta _2)\) as \((\mathcal {Y}_1\rtimes _{\eta _2} \mathcal {Y}_2,\theta (\eta _1,\eta _2))\) and denote it by \((\mathcal {Y}_1,\eta _1)\circ (\mathcal {Y}_2,\eta _2)\). Theorem 6.1 tells us that \((\mathcal {Y}_1,\eta _1)\circ (\mathcal {Y}_2,\eta _2)\) is an \((n,\ell )\)operator and that the composition of operators is associative. This allows us to define a new category: recall that \(\textrm{pMpx}^n\) denotes the class of all premaniplexes of rank n and let \({\bar{\textrm{pMpx}}}\) be the category whose objects are the classes \(\textrm{pMpx}^n\) with \(n \ge 1\), and whose arrows are voltage operators. An (n, m)operator is an arrow from \(\textrm{pMpx}^n\) to \(\textrm{pMpx}^m\) and the composition of arrows is defined as above. The neutral element at the object \(\textrm{pMpx}^n\) is the arrow \(({\textbf{1}}^n,\mu )\) where \(\mu \) is the mixing voltage, and the isomorphisms are precisely the voltage operators described in Example 3.1. It might be also interesting to study the analogous category obtained by considering rooted voltage operations.
Observe that the snub operation seems to act differently on orientable maniplexes than on nonorientable ones. However, the result of applying the snub operation to a nonorientable maniplex \(\mathcal {M}\) is one of the connected components of doing the same operation to the orientable double cover of \(\mathcal {M}\). This phenomenon is easy to understand with the following results:
Theorem 6.2
Let \(\mathcal {X}\) be an npremaniplex, \((\mathcal {Y}_1,\mu )\) be a mix noperator and \((\mathcal {Y}_2,\eta )\) be an (n, m)voltage operator with \(\mathcal {Y}_2\) connected. Let \(y_1\in \mathcal {Y}_1\) and \(y_2\in \mathcal {Y}_2\) be fixed and suppose that \(\eta (\Pi ^{y_2}(\mathcal {Y}_2))\) fixes \(y_1\). Then, the induced (colored) graph of \((\mathcal {X} \rtimes _\mu \mathcal {Y}_1)\rtimes _\eta \mathcal {Y}_2\) with vertex set \(\{((x,y_1),y):x\in \mathcal {X}, y\in \mathcal {Y}_2\}\) is isomorphic to \(\mathcal {X}\rtimes _\eta {\mathcal {Y}_2}\).
Proof
Because of Theorem 2.1, we may assume without loss of generality that \(\eta (W)\in \eta (\Pi ^{y_2}(\mathcal {Y}_2))\) for every path \(W\in \Pi (\mathcal {Y}_2)\), and thus \(\eta (W)\) fixes \(y_1\).
First we notice that the induced (colored) graph of \(\mathcal {Y}_1\rtimes _\eta {\mathcal {Y}_2}\) with vertex set \(\{(y_1,y):y\in \mathcal {Y}_2\}\) forms an isomorphic copy of \(\mathcal {Y}_2\). This is easy to see since the iadjacent flag to \((y_1,y)\) (in \(\mathcal {Y}_1\rtimes _\eta {\mathcal {Y}_2}\)) is \((\eta ({}^{i}y)y_1,y^i) = (y_1,y^i)\), because, by assumption, the voltages of paths in \(\mathcal {Y}_2\) fix \(y_1\).
Next we use Theorem 6.1 to see \((\mathcal {X} \rtimes _\mu \mathcal {Y}_1)\rtimes _\eta \mathcal {Y}_2\) as \(\mathcal {X}\rtimes _{\theta } (\mathcal {Y}_1\rtimes _{\eta \mathcal {Y}_2})\) with \(\theta =\theta (\eta ,\mu )\). This means that we can consider the iadjacent flag to \(((x,y_1),y)\) in \((\mathcal {X} \rtimes _\mu \mathcal {Y}_1)\rtimes _\eta \mathcal {Y}_2\) as the iadjacent flag to \((x,(y_1,y))\) in \(\mathcal {X}\rtimes _{\theta } (\mathcal {Y}_1\rtimes _\eta {\mathcal {Y}_2})\); we write \(((x,y_1),y)\leftrightarrow (x,(y_1,y))\) to denote that these two points are in correspondence under the isomorphism. On one hand, observe that by definition of \(\theta \), for any \(y\in \mathcal {Y}_2\) and any monodromy w, we have that \(\theta ({}^{w} (y_1,y))\) is \(\mu ({}^{{\eta ({}^{w} y})} y_1)\). On the other hand, by definition of \(\mu \) we have that \(\mu ({}^{{\eta ({}^{w} y})} y_1)=\eta ({}^{w} y)\).
Hence,
and since \(\big (\eta (^iy)x, (y_1,y^i)\big ) \leftrightarrow \big ((\eta (^iy)x, y_1),y^i\big )\), the theorem follows. \(\square \)
Corollary 6.3
Let \(\mathcal {X}\) be an npremaniplex, \((\mathcal {Y}_1,\mu )\) be a mix noperator and \((\mathcal {Y}_2,\eta )\) be an (n, m)voltage operator with \(\mathcal {Y}_2\) connected. Suppose that \(\eta (\Pi ^{y_2}(\mathcal {Y}_2)\) fixes every vertex of \(\mathcal {Y}_1\). Then \((\mathcal {X}\rtimes _\mu \mathcal {Y}_1)\rtimes _\mu {\mathcal {Y}_2}\) has a copy of \(\mathcal {X}\rtimes _\mu {\mathcal {Y}_2}\) for each vertex of \(\mathcal {Y}_1\).
We now have the tools to understand the relation between the snub of a nonorientable 3maniplex \(\mathcal {M}\) and the snub of its double cover. In Example 3.5 we saw that the orientable double cover of \(\mathcal {M}\) is \(\mathcal {M}\diamondsuit {\textbf{2}}^3_\emptyset =\mathcal {M}\rtimes _\mu {{\textbf{2}}^3_\emptyset }\). Hence, if \((\mathcal {Y},\eta )\) is the snub operator (see Fig. 4), then Corollary 6.3 tells us that \((\mathcal {M}\diamondsuit 2^3_\emptyset )\rtimes _{\eta } \mathcal {Y}\) consists of two copies of \(\mathcal {M}\rtimes _\eta \mathcal {Y}\). (Note that the voltages of \(\eta \) take values in \({{\,\textrm{Mon}\,}}^+(\mathcal {U}^3)\),the group that fixes the vertices of \({\textbf{2}}^3_\emptyset \).) In other words, the snub \(\mathcal {M}\) is an unrooted snub \(\mathcal {M}\diamondsuit {\textbf{2}}^3_\emptyset \).
The following is a similar example. Given a maniplex \(\mathcal {M}\) with a bipartite 1skeleton (that is, the graph with the 0faces as vertices and the 1faces as edges, with the induced incidence), one can find a voltage operator \((\mathcal {Y},\eta )\) such that \(\mathcal {M}\rtimes _{\eta }{\mathcal {Y}}\) has two connected components and in each component the vertices of one part of the bipartition are truncated while the ones in the other part remain unchanged. If now we apply this same operation to a maniplex \(\mathcal {M}'\) whose 1skeleton is not bipartite, the result is one of the connected components of \((\mathcal {M}'\diamondsuit {\textbf{2}}_{\{1,2,\ldots ,n1\}}^n)\rtimes _{\eta }{\mathcal {Y}}\).
7 Voltage Operations on the Universal Maniplex
Since a voltage operator \((\mathcal {Y},\eta )\) is a voltage premaniplex, it natural to consider the derived premaniplex \(\mathcal {Y}^\eta \). In this section we see that this derived premaniplex is in fact the one obtained from applying the corresponding voltage operation to the universal maniplex \(\mathcal {U}\). We shall use this fact to prove that Theorem 5.1 characterizes voltage operations.
Theorem 7.1
Let \(\mathcal {U}\) be the universal nmaniplex and let \((\mathcal {Y},\eta )\) a (n, m)voltage operator. Then \(\mathcal {Y}^\eta \) is isomorphic to \(\mathcal {U}\rtimes _{\eta }{\mathcal {Y}}\).
Proof
Let \(\Phi \) be a base flag of \(\mathcal {U}\). Consider the function \(\varphi : \mathcal {Y}^{\eta } \rightarrow \mathcal {U}\rtimes _{\eta }{\mathcal {Y}}\) given by \(\varphi (y,w) = (w \Phi , y)\). Recall that any flag of \(\mathcal {U}\) is of the form \(w \Phi \) for some (unique) \(w \in {{\,\textrm{Mon}\,}}(\mathcal {U})\), which implies that the function \(\varphi \) is bijective. Finally, observe that
\(\square \)
The proof of Theorem 7.1 is based on the fact that since \(\mathcal {U}\) is regular, its monodromy group acts regularly on its vertices (flags). We can generalize Theorem 7.1 as follows. Let \(\mathcal {X}\) be a regular premaniplex and let \((\mathcal {Y},\eta )\) be a voltage operator. If \(S_\mathcal {X}= \{w \in {{\,\textrm{Mon}\,}}(\mathcal {U}): w x= x \ \textrm{for} \ \textrm{all} \ x\in \mathcal {X}\}\), that is, the kernel of the projection \({{\,\textrm{Mon}\,}}(\mathcal {U}) \rightarrow {{\,\textrm{Mon}\,}}(\mathcal {X})\), then \({{\,\textrm{Mon}\,}}(\mathcal {X}) \cong {{\,\textrm{Mon}\,}}(\mathcal {U})/ S_{\mathcal {X}}\). Let \(\eta _\mathcal {X}:\Pi (\mathcal {Y})\rightarrow {{\,\textrm{Mon}\,}}(\mathcal {X})\) be the voltage assignment on \(\mathcal {Y}\) we get by reducing \(\eta \) to \({{\,\textrm{Mon}\,}}(\mathcal {X})\), that is \(\eta _\mathcal {X}(W):=\eta (W) S_\mathcal {X}\). Then by using the exact same argument as in Theorem 7.1 we can see that \(\mathcal {X}\rtimes _{\eta \chi }{\mathcal {Y}}\) is isomorphic to \(\mathcal {Y}^{\eta _\mathcal {X}}\).
An immediate consequence of Theorems 5.1 and 7.1 is that operators \((\mathcal {Y}, \eta _{1})\) and \((\mathcal {Y},\eta _{2})\) with \(\eta _{1}\) and \(\eta _{2}\) equivalent voltages (that is, with isomorphic derived graphs as in Equation ((2.1))) yield equivalent voltage operations. More precisely,
Proposition 7.2
Let \((\mathcal {Y},\eta _{1})\) and \((\mathcal {Y},\eta _{2})\) be two (n, m)operators with equivalent voltages \(\eta _{1}\) and \(\eta _{2}\). Then for every npremaniplex \(\mathcal {X}\),
Moreover, there is an isomorphism \(\mathcal {X}\rtimes _{\eta _{1}} \mathcal {Y}\rightarrow \mathcal {X}\rtimes _{\eta _{2}} \mathcal {Y}\) such that the following diagram commutes:
Proof
From [12] we know that \(\mathcal {X}\cong \mathcal {U}/ \Gamma \) for certain \(\Gamma \le \Gamma (\mathcal {U})\), now
If we start with a vertex (x, y) in \(\mathcal {X}\rtimes _{\eta _1}\mathcal {Y}\) and apply the natural isomorphisms between consecutive terms in the equation above, we get the following sequence, where \(\Phi \) is a base flag of \(\mathcal {U}\):
We can see that this isomorphism makes the diagram in Equation ((7.1)) commute. \(\square \)
In the proof of Proposition 7.2 we strongly used that if \(\mathcal {X}\cong \mathcal {U}/\Gamma \) for \(\Gamma \le \Gamma (\mathcal {U})\), then \(\mathcal {X}\rtimes _\eta \mathcal {Y}\cong (\mathcal {U}\rtimes _\eta \mathcal {Y})/\Gamma \) for any voltage operator \((\mathcal {Y},\eta )\) (which is Theorem 5.1 of this paper). In the following result we will show that this property characterizes all voltage operations.
Theorem 7.3
Let \({{\,\mathrm{\mathcal {O}}\,}}\) be a mapping that assigns an mpremaniplex \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {X})\) to each npremaniplex \(\mathcal {X}\). Assume that there is an action of \(\Gamma (\mathcal {U})\) on \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {U})\) such that \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {U}/ \Gamma ) \cong {{\,\mathrm{\mathcal {O}}\,}}(\mathcal {U})/\Gamma \) for every \(\Gamma \le \Gamma (\mathcal {U})\), then there exists an (n, m)voltage operator \((\mathcal {Y}, \eta )\) such that
for every premaniplex \(\mathcal {X}\).
Proof
Let \({\textbf{1}}^{n}\) denote the unique npremaniplex with one vertex and define \(\mathcal {Y}\) as the premaniplex \({{\,\mathrm{\mathcal {O}}\,}}({\textbf{1}}^{n})\). Observe that
which implies that there exists a voltage assignment \(\eta :\Pi (\mathcal {Y}) \rightarrow \Gamma (\mathcal {U}) \cong {{\,\textrm{Mon}\,}}(\mathcal {U})\) such that \(\mathcal {Y}^{\eta } \cong {{\,\mathrm{\mathcal {O}}\,}}(\mathcal {U})\). The pair \((\mathcal {Y}, \eta )\) defines a voltage operator and by Theorem 7.1, \(\mathcal {U}\rtimes _\eta \mathcal {Y}\cong \mathcal {Y}^{\eta } \cong {{\,\mathrm{\mathcal {O}}\,}}(\mathcal {U}) \).
Finally, if \(\mathcal {X}\) is a premaniplex and \(\Gamma \le \Gamma (\mathcal {U})\) is such that \(\mathcal {X}\cong \mathcal {U}/\Gamma \), then
\(\square \)
We often come across operations that are well defined for some family F of premaniplexes (for example, for convex polytopes) but such that it is not immediately evident how to generalize them for all premaniplexes. One may ask if it is possible to extend operations of this kind to all premaniplexes (of the given rank) in such a way that the operation is a voltage operation. As an example, in Sect. 4, we have found voltage operations that extend the Wythoffian operations, the kbubble and the trapezotope operation to all premaniplexes. Theorem 7.3 and Corollary 5.6 answer this question and find the corresponding voltage operation, when possible. The idea is as follows: suppose there is some regular premaniplex \(\mathcal {P}\) and an operation \({{\,\mathrm{\mathcal {O}}\,}}\) such that we already know \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {P})\). According to the proof of Theorem 7.3, if \({{\,\mathrm{\mathcal {O}}\,}}\) is indeed a voltage operation \(\mathcal {X}\mapsto \mathcal {X}\rtimes _\eta \mathcal {Y}\), then \(\mathcal {Y}\) should be \({{\,\mathrm{\mathcal {O}}\,}}({\textbf{1}}^n)\), but due to Theorem 5.1\(\mathcal {Y}\) must coincide with \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {P})/\Gamma (\mathcal {P})\). Now let \(\nu :\Pi (\mathcal {Y})\rightarrow \Gamma (\mathcal {P})\) be the voltage assignment such that \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {P})\) is isomorphic to \(\mathcal {Y}^\nu \). Corollary 5.6 tells us that \(\nu \) is obtained by replacing each occurrence of \(r_i\) by \(\rho _i\) in the voltage assignment \(\eta :\Pi (\mathcal {Y})\rightarrow {{\,\textrm{Mon}\,}}(\mathcal {U})\). So we can recover \(\eta \) by replacing every instance of \(\rho _i\) by \(r_i\) in \(\nu \). Note that, in general, such replacement is not well defined unless \(\mathcal {P}\) is the universal polytope \(\mathcal {U}\), however with some intuition we can find the right way to do it for the natural occurring operations. However, observe that one can do this without knowing if \({{\,\mathrm{\mathcal {O}}\,}}\) was in fact a voltage operation. To know if \({{\,\mathrm{\mathcal {O}}\,}}\) is a voltage operation or not, one must see that \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {X})=\mathcal {X}\rtimes _\eta \mathcal {Y}\) for \(\mathcal {X}\) in the family F. In fact, if for some \(\mathcal {X}\in F\) we observe that \({{\,\mathrm{\mathcal {O}}\,}}(\mathcal {X})\ne \mathcal {X}\rtimes _\eta \mathcal {Y}\), then \({{\,\mathrm{\mathcal {O}}\,}}\) cannot be seen as a voltage operation.
8 Final Remarks and Open Problems
We have seen that voltage operations generalize classical operations on maps and polytopes and allow us to define such classical operations on premaniplexes.
One can see that voltage operations naturally generalize to hypertopes (thin residually connected geometries), “complexes”, and their quotients. Following [32], a complex is a properly nedge colored nvalent graph (note that a complex is a combinatorial map in the sense of Vince [30]). Complexes generalize (the “chamber graphs” of) hypertopes. Thus, one can define a precomplex as a nvalent pregraph that has been properly nedge colored. In this context, the definition of a voltage operator as well as the results in this paper hold if we ask \(\eta :\Pi (\mathcal {Y})\rightarrow W^n\), where \(W^n\) is the group generated by n involutions with no other relations among the generators. In particular, this allows us to define operations like the ones given in Sects. 3 and 4 to hypertopes.
Symmetry type graphs and voltage assignments have proven to be strong tools to study polytopes and maniplexes. A natural (and wellknown) problem that arises when dealing with symmetry type graphs is the following:
Problem 1
Given a premaniplex \(\mathcal {T}\), does there exist a polytope (or maniplex) such that its symmetry type graph (with respect to the full automorphism group) is \(\mathcal {T}\)?
A particular example of this problem was the question posted in the early 1990’s by Schulte and Weiss of whether or not there exist chiral polytopes of all ranks (which was solved by Pellicer in 2010 [25]). Of course, one can generalize Problem 1 to hypertopes.
Problem 2
Given a precomplex \(\mathcal {T}\), does there exist a hypertope (or complex) such that its symmetry type graph (with respect to the full automorphism group) is \(\mathcal {T}\)?
One thing that one tries to do when dealing the above question is to use voltage assignments on \(\mathcal {T}\) to lift to a maniplex \(\mathcal {M}\). In [22] the voltage assignments that give a polytope as the derived graph are characterized. Of course the voltage group acts by automorphisms on \(\mathcal {M}\) and the quotient of \(\mathcal {M}\) by it is precisely \(\mathcal {T}\). However, \(\mathcal {M}\) can have (and often does) symmetries not coming from the voltage group. In the context of voltage operations, we have shown that given a voltage operator \((\mathcal {Y}, \eta )\) and a premaniplex \(\mathcal {X}\), all automorphisms of \(\mathcal {X}\) act as automorphisms of \(\mathcal {X}\rtimes _{\eta }\mathcal {Y}\). However, again, \(\mathcal {X}\rtimes _{\eta }\mathcal {Y}\) might have extra symmetry (for example, in the case when one applies the medial operation to a selfdual map). It is natural to ask when is it true that automorphisms of \(\mathcal {Y}\) also lift to \(\mathcal {X}\rtimes _{\eta }\mathcal {Y}\), and when all automorphisms of \(\mathcal {X}\rtimes _{\eta }\mathcal {Y}\) come either from \(\mathcal {X}\) or from \((\mathcal {Y}, \eta )\). We refer to [15] for more details about these questions. Answering, at least partially, these questions might be of great help for solving Problem 1 (at least partially).
Problem 3
Give necessary conditions on \((\mathcal {Y},\eta )\) so that one can compute the symmetry type graph of \(\mathcal {X}\rtimes _{\eta }\mathcal {Y}\) with respect to its full automorphism group in terms of \(\mathcal {X}\) and \((\mathcal {Y},\eta )\).
If one’s interest is in polytopes rather that in (pre)maniplexes, one can ask when a voltage operation preserves polytopality, though we might not care if the result of the operation is not connected. More precisely,
Problem 4
Give necessary and sufficient conditions on \((\mathcal {Y}, \eta )\) for \((\mathcal {P},\Phi )\rtimes _\eta (\mathcal {Y},y)\) to be a polytope, for all rooted polytopes \((\mathcal {P}, \Phi )\) and \(y\in \mathcal {Y}\).
Similarly,
Problem 5
Give necessary and sufficient conditions on \((\mathcal {Y}, \eta )\) for \(({{\mathcal {H}}},\Phi )\rtimes _\eta (\mathcal {Y},y)\) to be a hypertope, for all rooted hypertopes \(({{\mathcal {H}}}, \Phi )\) and \(y\in \mathcal {Y}\).
Notes
Usually a voltage assignment \(\xi \) is defined such that \(\xi (W_1 W_2)=\xi (W_1)\xi (W_2)\) (cf. [19]). The reason we do it the other way is because we are considering right actions, as is customary in the polytopes and maniplexes literature.
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Acknowledgements
The authors thank the anonymous referee for the helpful corrections. They also thank the financial support of CONACyT grant A1S21678. This paper was completed while the second author held a Post Doctoral Research Associate position in the Department of Mathematics at Northeastern University and the third author was supported by the Post Doctoral Scholarship Program at UNAM (DGAPA), Mexico.
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Hubard, I., Mochán, E. & Montero, A. Voltage Operations on Maniplexes, Polytopes and Maps. Combinatorica 43, 385–420 (2023). https://doi.org/10.1007/s00493023000187
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DOI: https://doi.org/10.1007/s00493023000187